Integrating hesitant multiplicative preferences and best‐worst method into the graph model for conflict resolution with applications to water resources conflict in the Yellow River Basin

1 Introduction

Water resource conflicts often stem from the uneven distribution of resources and a surge in water demand, leading to intensified competition between different regions and industries (Li et al., 2021). Due to the scarcity of water resources, it is difficult to meet the water needs of agriculture, industry, and residents, thus affecting economic production and quality of life (Gu et al., 2023). In addition, the excessive development and poor management of water resources have led to ecological degradation, which poses a serious threat to the ecological environment of water source areas and reduces biodiversity (Wang et al., 2025). Ultimately, the competition for water resources intensifies social conflicts, which may lead to cross-regional and inter-departmental conflicts, threatening social stability and sustainable development (Peng et al., 2023; Yuan et al., 2023). The Yellow River is considered the “mother river” of China, and its basin is the center of China’s politics, economy, and culture. Water resource conflicts in the Yellow River Basin not only damage the ecological environment and affect agricultural and residential water use, but they also trigger social conflict between different provinces. The conventional approach to resolving water resource conflicts is based on environmental and engineering perspectives, thus overlooking decision-making behavior and interaction among various decision-makers (DMs) involved in an uncertain conflict environment (Wang et al., 2022b; Wang et al., 2024b; Li et al., 2025; Mukherjee et al., 2025). The uncertainty of DMs’ preference cognition and the complexity of conflict exacerbate the difficulty of conflict resolution. In this study, various rational policy suggestions are proposed to resolve water resource conflicts from a management perspective by combining two classic decision analysis methods with uncertainty preferences (Tang et al., 2025). It fully safeguards the interests and demands of each DM and maximizes consensus on decision-making.

Conflicts, including over water resources, are a social phenomenon of opposition caused by differences in the distribution of interests among different DMs (Zhao et al., 2019; Sabino and Rêgo, 2023; Zhang et al., 2024). To provide constructive suggestions for each DM in conflicts, classical game theory and its different expansion methods have been proposed and expanded based on both mathematics and management theory. However, these methods depend on clear data. To address conflicts more quickly and in a standardized manner, a novel and practical approach called the “graph model for conflict resolution” (GMCR) (Kilgour et al., 1987) was first designed based on both quantitative and qualitative analysis and was subsequently enriched and developed by many globally renowned researchers (Wang et al., 2022a; 2023a; b; Wang et al., 2024a). As the name suggests, GMCR only needs to draw directed graphs of conflicts based on less relative preference information rather than a large amount of precise quantitative information for carrying out the model and analysis process. This advantage lays the theoretical foundation for GMCR to resolve conflicts by using preference relation theory and graph theory. Considering its advantages such as a wide application field, limited data usage, and simple structure, conflicts in multiple fields of real-life society can be resolved by using GMCR. Wu et al. (2023) constructed the hesitant fuzzy GMCR to resolve Lancang–Mekong Basin cross-country water resource conflicts. Wang et al. (2024a) first proposed GMCR under an information asymmetry and applied it to resolve Niangziguan Springs Basin water resource conflicts. These studies have shown that GMCR has significant advantages in analyzing water resource conflicts in different river basins. Ningxia Province in China is the only province in the country that belongs to the Yellow River Basin. Water resource sustainable development by the Ningxia government promotes industrial revitalization and improves green development (Shi et al., 2024). However, to pursue maximum benefits, enterprises may encounter environmental problems such as substandard wastewater and groundwater pollution during the production process, which inevitably conflict with local governments and residents. Thus, the complex social background and high social demonstration significance of water resource conflicts in the Yellow River Basin make it an ideal choice for our research case.

As an interesting perspective in the basic theoretical research of the GMCR, the characterization of different DMs’ preference types and their corresponding stability have received widespread attention from researchers. Initially, crisp preference (≻, ∼) is used to represent a DM’s relative preference in GMCR. To describe a DM’s uncertainty preferences in GMCR, researchers have successfully developed and enriched various types of uncertain preferences, starting with unknown preferences (Li et al., 2004) and different types of the same (Bashar et al., 2012; Kuang et al., 2015; Wu et al., 2019; Wu et al., 2021; Wu et al., 2024). Hesitant multiplicative preference relations (HMPRs) have received attention from researchers since they were proposed. Xia and Xu (2013) initially proposed HMPRs to describe the uncertain preferences of DMs by using a hesitant multiplicative set (HMS), represented by Saaty’s ratio scale [1/9, 9]. There is now a large body of research regarding HMPRs. For example, two novel solving models have been constructed by Zhang and Wu (2014) to calculate weights from HMPRs. Liu et al. (2020) first converted HMPRs into generalized HMPRs by using transformation formulas. DMs therefore tend to use HMPRs to show their self-preference degree data rather than specific preference relations data when faced with a complex decision-making problem in uncertain environments. HMPRs do not yet seem to have been considered in the paradigm of GMCR. HMPRs use Saaty’s ratio scale [1/9,9] to more intuitively express DMs’ hesitation and uncertainty; compared with traditional fuzzy preferences, uncertainty expression is more accurate. Thus, this study first uses the HMPRs to characterize DM preferences in GMCR to ensure the authenticity and scientific integrity of preference expression. GMCR with HMPRs uses the preference expression theory to address uncertainty generated from cognitive judgment, which ensures that stability analysis results are more reliable and persuasive.

Three existing preference ranking methods in GMCR cannot be used to rank all states in uncertainty environments and are only applied for a single DM. For example, when a composite DM (CDMs) in GMCR consists of a group of individual DMs, the existing preference ranking methods in this situation cannot apply (Wu et al., 2023). Note that CDMs indicate that an involver is composed by a set of single DMs with the same interests. The heterogeneity among CDMs is an interesting direction worth studying, but it is not the focus of this study. Wang et al. (2024b) studied the heterogeneity among decision makers under the graph model framework, as well as the application of environmental conflicts. As is well known, the multi-criteria decision-making method applies various mathematical theories to determine weights of criteria to rank alternatives in complex environments. Recently, the best–worst method (BWM) has received widespread attention (Rezaei, 2015; 2016; Mou et al., 2017; Mi and Liao, 2019; Xu et al., 2021; Xu and Wang, 2024). BWM needs only 2n−3 comparisons when making a judgment between any two alternatives, meaning that it can effectively ensure logical rationality and apply in many management decision problems (Rezaei et al., 2018; Ye et al., 2024). To facilitate the reader’s understanding, different criteria in BWM are equivalent to all states in GMCR. It is evident that there is much research on BWM, but none has considered it within the framework of GMCR. More importantly, each DM needs to consider various criteria for a comprehensive ranking result to resolve the problems generated by subjective judgment and produce more realistic and reliable results. Therefore, considering the validity and scientific integrity of HMPRs in characterizing uncertain preferences, this study first constructs BWM with HMPRs (HMBWM). In group decision-making, HMBWM is only used to determine the preference ranking between different alternatives in uncertain environments, while in GMCR, HMBWM is only considered to determine DM preferences, provide a prerequisite for conflict analysis, and does not need to consider consistency. The innovation of this study is that HMBWM in GMCR is used to rank all states, which can alleviate the adverse effects of information loss or information uncertainty to the greatest extent possible.

Stability concepts in GMCR are defined as finding one or several states among numerous states that can be accepted by stakeholders (Nash, 1950; Nash, 1951). Furthermore, a global equilibrium situation to the conflicts means that the state is stable for each DM. As the types of DM preference relations studied within the framework of GMCR become increasingly diverse, the corresponding stability definitions have also been further perfected. The literature suggests that various types of stability definitions have been constructed, including Maximin _*h* stability (Rêgo and Vieira, 2020) and optimism pessimism stability (Sabino and Rêgo, 2023). The traditional stability definitions in original GMCR were only used to obtain stabilizing solutions for certain conflicts with crisp preferences, so researchers are required to expand appropriate stability definitions based on different complex conflict environments. Thus, hesitant multiplicative stability concepts are first proposed and used to obtain an optimal solution of water resource conflicts in the Yellow River Basin. Hence, the stability analysis result obtained from multiplicative stability concepts is more convincing and accurate than traditional GMCR in resolving conflicts under uncertain environments. The contributions of this study are as follows.

• To better analyze conflicts in uncertainty, HMS may better align with DM preference expression customs. HMPRs can be used to overcome the ambiguity of DM’s subjective evaluations. Furthermore, HMBWM is constructed to rank all states for each DM with uncertainty preferences.
• A set of new stability definitions with HMPRs has been proposed in GMCR with uncertain environments. It proposes a new skill for resolving conflicts in uncertain environments.
• The complex reality of water resource conflicts in the Yellow River Basin indirectly illustrates that DMs’ uncertainty preferences have a real social background, and it also demonstrates the effectiveness and urgency of the proposed model.

The remainder of this article is organized as follows. Section 2 details existing knowledge of GMCR and the nature of HMPRs. In Section 3, an optimization model is proposed in HMBWM. In Section 4, HMPRs are integrated into GMCR to propose a set of new hesitant multiplicative stability definitions. Section 5 presents hesitant multiplicative stability analysis of water resource conflicts in the Yellow River Basin. In Section 6, comparative analysis is given. Finally, conclusions, limitations, and future perspectives are outlined by Section 7.

2 Preliminary

In this section, in order to facilitate readers’ understanding of the methods used in this article, basic knowledge about and the logical framework of GMCR and HMPRs are introduced. This study combines traditional GMCR with HMPRs and extends them to uncertain environments to generate new conflict analysis methods.

2.1 GMCR

Conflict analysis methods are being enriched and developed as conflicts become increasingly severe and complex, with GMCR determined according to traditional game theory. GMCR is composed of DMs, feasible states, a directed graph, and personal preferences, represented as GMCR=<N, S, G, P>. Below, we explain the meaning of each element of GMCR in detail.

• N = {1, 2, …, n} (N ≥ 2) is stakeholders. When describing the background of real-life conflicts, the primary task is to identify all their participants or stakeholders. That is to say, individuals who can independently express personal preference judgments during the process of conflict evaluation or a coalition include different DMs with the same wishes and aims.
• S = {s ₁, s ₂, …, s _*m*} is feasible states. The selection of each strategy is either “Y” or “N” for each DM.
• G is the DMs’ direct graph. Each feasible state in GMCR is represented by all vertices in a direct graph of conflicts. All possible transition cases are directly represented by using directed arcs in a directed graph. (s, s ₁)∈A _*k* is directed arcs.
• P = {∼_*k*, ≻_*k*} represents a DM’s preference degree, “∼_*k*” represents that one has same preferences between two states, and “≻_*k*” indicates that one prefers one state over another state.

Furthermore, if the given state is the optimal solution for all DMs in GMCR and it is considered as the global stability state or equilibrium state of conflicts, the specific solving and analysis process is shown in Figure 1.

Definition 1 Within the traditional GMCR, R k s = s 1 ∈ S , s , s 1 ∈ A k is used to represent the unilateral reachable list.

Definition 2 Within the GMCR, R k + s = s 1 ∈ R k s , s 1 ≻ k s is used to represent the unilateral improved reachable set.

Next, when conflicts involve more than two DMs, H⊆N is used to represent a coalition of DMs, and R _*H*(s) and R H + (s) are defined by using an iterative method according to Definitions 1 and 2 (Wang et al., 2023b; Wang et al., 2024a). In particular, Ω_*H* (s, s _*i*) represents a set of stakeholders that exist in the determining process of R _*H*(s). Similarly, the acquisition process of Ω H + s , s i is similar to the Ω_*H* (s, s _*i*).

• if k ∈ H and s 1 ∈ R k s, and there exists s 1 ∈ R H s and k ∈ Ω H s , s 1;
• if k ∈ H, s 1 ∈ R H s, and s 2 ∈ R k s 1, then we obtain Ω H s , s 1 ≠ k, s 2 ∈ R H s, and k ∈ Ω H s , s 2.

Definition 3 In the GMCR, the steps to obtain R _*H*(s) are shown in the following:

It is evident that the acquisition of R H + (s) is very similar to Definition 3. The difference between R _*H*(s) and R H + (s) is that the movement in R H + (s) is an improvement. Furthermore, Definitions 4−7 are used to display the basic definitions in GMCR.

Definition 4 Nash stability (Nash, R) in original GMCR means that there exist R k + s = ∅ for DM k at state s.

Definition 5 General metarationality (GMR) in original GMCR represents that, for each state s 1 ∈ R k + s, there is at least one state s 2 ∈ R N − k s 1 that satisfies s 2 ≺ ∼ s.

Definition 6 Symmetric metarationality (SMR) in original GMCR means that, for each state s 1 ∈ R k + s, there is at least one state s 2 ∈ R N − k s 1 that satisfies s 2 ≺ ∼ s and s 3 ≺ ∼ s for all state s 3 ∈ R k s 2.

Definition 7 Sequential stability (SEQ) represents, in original GMCR, that for each state s 1 ∈ R k + s there is at least one state s 2 ∈ R N − k + s 1 that satisfies s 2 ≺ ∼ s.

2.2 HMPRs

In particular, it is supposed that a _*ij* is valued by using the 1–9 ratio scale.

The fuzzy sets were first proposed to describe the uncertain situation by Zadeh (1965). Many expansion forms of fuzzy sets have been designed to resolve various complex uncertain decision-making situations; an important concept is called “hesitant fuzzy sets” (HFSs) (Torra, 2010).

a i j = 1 a j i , a i i = 1 , a i j ∈ 0 , + ∞ , ∀ i , j = 1 , 2 , … , m .

E = x , h E x | x ∈ X .

Definition 8 (Saaty, 1977). Let X = {x ₁, x ₂, …, x _*m*} be a set of criteria. A=(a _*ij*)_*m*×*m* is then considered as multiplicative preference relations over X × X if and only if (iff) the elements in matrix A satisfy the following conditions:

Definition 9 (Torra, 2010). The HFSs over X is represented by a function when X equals a subset of [0, 1]. There then exists

However, a 0.1–0.9 scale is sometime not in line with the intuitive situation in real-life decision making. In other words, the intuitive judgment of DMs is a reality that must be considered in negotiation processes. Thus, to overcome this shortcoming, according to the advantages of HFSs and multiplicative preference relations, the symmetrical 0.1–0.9 scale in HFSs was changed to an asymmetrical 1–9 scale by Xia and Xu (2013), which can be used to character membership degree.

M = x , h M x | x ∈ X .

Definition 10 (Xia and Xu, 2013). The HMS over X is represented by a function that, when considered to Xi, equals a subset of [1/9, 9]:

For convenience, h _*M*(x) is abbreviated as h(x), and it is named hesitant multiplicative element (HME) in HMS. All HMEs are represented as M. Xia and Xu (2013) first defined HMPRs as follows.where all elements in HME h _*ij* are considered the increase from left to right, h i j σ t is tth smallest preference degree, and #h _*ij* is the total number. Note that h _*ij* is used to represent the uncertain preference between the alternatives x _*i* and x _*j*.

h i j σ t h j i σ l h i j − t + 1 = 1 , hii = 1 , # hij = # hji , i , j = 1 , 2 , … , m ,

Definition 11 The HMPRs H over X are defined as H=(h _*ij*)_*m*×*mX* × X, where h _*ij* = {h i j t, t = 1, 2, … , #h _*ij*} is an HME, and it includes all preference expression values between two criteria x _*i* and x _*j*. Then, h _*ij* possesses the following nature:

3 Novel preference ranking method in GMCR: HMBWM

The traditional graph model is based on the simple preferences of DMs, which results in the preference acquisition methods of traditional graph models being limited to solving simple preferences and is difficult to apply to complex and uncertain environments. This defect greatly limits the application scope of graph models. Therefore, we propose a new preference acquisition method in the graph model framework by combining the BWM method and HMPRs. Rezaei (2015); Rezaei (2016) first proposed the BWM and its solving steps. However, the traditional BWM cannot resolve decision making problems in uncertain environments. Thus, considering that the pairwise comparisons in decision making problems are represented by HMS, a novel optimization model is thereby constructed in the HMBWM model. First, there exist two new vectors,where HMBO is hesitant multiplicative best-to-others, HMOW is hesitant multiplicative others-to-worst vector, and h B B = h W W = 1.

H M B O = h B 1 , h B 2 , … , h B j , … , h B m , H M O W = h 1 W , h 2 W , … , h j W , … , h m W T , (1)

We here propose the function of the weighted geometric mean of HME, where different values can be calculated into one value:

g H M S = # h u j ∏ t = 1 # h i j h i j t ​ . (2)

According to Equation 2, the most possible representative preference values can be easily obtained in both HMBO and HMOW vectors:

g H M B O = g h B 1 , g h B 2 , … , g h B j , … , g h B m , (3)

g H M O W = g h 1 W , g h 2 W , … , g h j W , … , g h m W T . (4)

As for the g (HMBO) and g (HMOW) vectors, the multiplicative consistent relations between each component of Equation 3 and Equation 4 and the weights can be described as follows:

w B w j = g h B j , w j w W = g h j W .

To calculate the rational weights from the preference vector determined by each DM, the maximum absolute differences w B / w j − g h B j and w j / w w − g h j W for all j should be as small as possible.

Model 1.

min ξ g s . t . w B w j − g h B j ≤ ξ g , j = 1 , 2 , . . . , m w j w W − g h j W ≤ ξ g , j = 1 , 2 , . . . , m ∑ j = 1 m w j = 1 w j ≥ 0 , j = 1 , 2 , . . . , m . (5)

According to Model 10, the calculating weight results can be used to rank the criteria. The calculating weights over all alternatives can be used to rank all states in GMCR. More importantly, HMBWM can be considered a new skill to obtain real DM preferences in GMCR, which provokes consideration of other research methods in GMCR structures to expand research ideas.

Some necessary principles are established to establish a close connection between BWM and GMCR. Thus, we first define a new concept of ordinal weight vector to character DM preference ranking by using weights determined by HMBWM.

Definition 12 The ordinal weight vector for DM k in GMCR with HMPRs is defined as η k = η k 1 , η k 2 , . . . , η k m, where η k i, 1 ≤ i ≤ m is the corresponding ordinal weight value of the s _*i*, and η k i is any sole integer in [1, m], represented as η k s i = η k i. Thus, the larger the ordinal weight value of v _*i*, the higher the preference of the corresponding state s _*i*.

Finally, in GMCR with HMPRs, HMBWM provides a new technology to rank all states. Compared with these existing ranking methods, HMBWM is simpler, more effective, and has a wider range of applications. The HMBWM method proposed within the framework of GMCR provides a feasible tool for solving the preferences of DMs in complex and uncertain environments.

4 GMCR with HMPRs

The four basic stability definitions within the traditional GMCR framework provide theoretical bases for solving conflicts. However, considering the conflict analysis graph model in complex and uncertain environments, these definitions need to be appropriately expanded to address real-world situations.

4.1 Underlying definitions of GMCR with HMPRs

Similar to Definitions 1 and 2 in the original GMCR, two underlying definitions have been defined to construct GMCR with HMPRs.

H M R k s = s 1 ∈ S , s , s 1 ∈ A k .

H M R k + s = s 1 ∈ H M R k s , s 1 ≻ k s .

• if k ∈ H and s 1 ∈ H M R k s, and there exist s 1 ∈ H M R H s and k ∈ Ω H s , s 1;
• if k ∈ H, s 1 ∈ H M R H s, and s 2 ∈ H M R k s 1, then we obtain Ω H s , s 1 ≠ k, s 2 ∈ H M R H s, and k ∈ Ω H s , s 2.

Definition 13 The hesitant multiplicative unilateral reachable (HMR) list in GMCR with HMPRs is

Definition 14 The hesitant multiplicative unilateral improvement list (HMUIL) in GMCR with HMPRs is

Definition 15 The steps to obtain a reachable list HMR _*H*(s) in GMCR with HMPRs are shown as follows:

Definition 16 The steps to obtain a reachable list H M R H + s in GMCR with HMPRs can be obtained as described in Definition 16.

4.2 Hesitant multiplicative stability definition of GMCR

Definition 17 A given state s in enhanced GMCR is hesitant multiplicative Nash (HMNash) for DM k iff H M R k + s = ∅.

Definition 18 A given state s in enhanced GMCR is hesitant multiplicative general metarational (HMGMR) for DM k iff s ₁∈ H M R k + s, there exists another state s ₂∈ H M R N − k s 1 and has s≻_*k* s ₂.

Definition 19 A given state s in enhanced GMCR is hesitant multiplicative symmetric metarational (HMSMR) for DM k iff s ₁∈ H M R k + s, there exists another state s ₂∈ H M R N − k s 1 and has s≻_*k* s ₂, and for any state of s ₃∈HMR _*k*(s ₂) and has s≻_*k* s ₃.

Definition 20 A state s in enhanced GMCR is hesitant multiplicative sequential stability (HMSEQ) for DM k iff s ₁∈ H M R k + s, there exists another state s ₂∈ H M R N − k + s 1 and has s≻_*k* s ₂.

Thus, Definitions 17−20 are the four basic stability definitions in GMCR with HMPRs. Those new proposed definitions are designed based on Definitions 4−7. The main difference between them is that new proposed definitions are applied in an uncertainty decision-making environment. Definitions 17−20 can accurately depict the interest exchange between DMs and analyze the development trend of conflicts.

Recently, Rêgo and Vieira (2017) explored symmetric sequential stability (SSEQ), which provides more information regarding the stability of states for analysts. We extend original SSEQ to the HMPRs environment—“hesitant multiplicative symmetric sequential stability” (HMSSEQ).

Definition 21 A state s in enhanced GMCR is HMSSEQ for DM k iff s ₁∈ H M R k + s, there exists another state s ₂∈ H M R N − k + s 1 and possesses s≻_*k* s ₂, and for any state of s ₃∈HMR _*k*(s ₂) and possesses s≻_*k* s ₃.

Obviously, HMSSEQ is developed from Definition 19. HMSSEQ is also a weak stability definition, and it has three-step distance movement. The main difference is that the opponent DMs of focus DM k in HMSSSEQ only consider the rational sanctions. The logical relation among Definitions 17−21 is shown in Theorem 1.

S k H M N a s h ⊆ S k H M S S E Q ⊆ S k H M S M R ⊆ S k H M G M R ,

S k H M N a s h ⊆ S k H M S S E Q ⊆ S k H M S E Q ⊆ S k H M G M R .

Theorem 1 Within GMCR with HMPRs, the relationships among hesitant multiplicative stability definitions are described as follows:

Proof. First, HMNash stable means that H M R k + s = ∅, representing that s is HMSSEQ. Second, according to HMSSEQ, there exists for s ₁∈ H M R k + s, there exists s ₂∈ H M R N − k s 1 and has s≻_*k* s ₂, and for any state of s ₃∈HMR _*k* (s ₂) and s≻_*k* s ₃, which represents that s is HMSMR stable. Furthermore, we obtain that for each state s ₁∈ H M R k + s, there exists s ₂∈ H M R N − k s 1 and possesses s≻_*k* s ₂. The above situation shows that s is HMGMR stable. Thus, logical relationships (13) and (14) can be obtained. The result is shown in Figure 2.

Finally, in order to facilitate readers’ understanding of the newly proposed conflict analysis method GMCR with HMPRs and how it can be applied to solve real-world conflicts, a pseudocode description for reference and reproduction is shown in Table 1.

TABLE 1. Pseudocode description of GMCR with HMPRs.

5 Application to water resource conflicts in the Yellow River Basin

To demonstrate the effectiveness and feasibility of the newly proposed GMCR with HMPRs, we apply it to real-life water resource conflicts in the Yellow River Basin.

5.1 Backgrounds

The ecology of the Yellow River Basin is extremely fragile, with frequent natural disasters. On the one hand, due to various factors such as climate and human activities, its ecosystem has been severely damaged, with serious problems such as forest degradation, desertification, soil erosion, ecological imbalance, and frequent natural disasters such as floods, which have had a serious impact on the life of local people. On the other hand, in the process of large-scale urbanization and agricultural development in the Yellow River Basin, high-quality development is insufficient, highly polluting enterprises are concentrated, and industrial enterprises cause serious pollution and damage to the local ecological environment, greatly hindering the sustainable development of the local economy and society. Ningxia Province is one of the five major autonomous regions in China. It has a small geographical area, but it faces typical problems such as ecological fragility, water resources scarcity, and many challenges in industrial transformation and upgrading in the Yellow River Basin (Figure 3).

The chemical industry is economically large, has high correlation and a strong driving force, and is an important component of the national economy and the industrial economy of the autonomous region, having developed there rapidly. The chemical industry mainly comprises coal chemical, petrochemical, fertilizer, and chemical new materials. However, with the deepening of industrialization and the increase in the number of industrial enterprises, industrial wastewater discharge has become a major challenge facing the ecology of rivers in China. Once harmful substances in industrial wastewater are discharged into rivers, they not only endanger the survival of aquatic organisms but also threaten the living environments of humans. In 2021, the Central Ecological and Environmental Protection Inspection Team discovered that a large number of enterprises in the Fine Chemical Industry Park in Pingluo County, Ningxia Province faced problems such as the illegal utilization of Yellow River water resources and significant pollution hazards. The channel for the discharge of chemical industry wastewater into the Yellow River Basin and its black sludge are shown in Figure 4. The local protection bureau and the government immediately took relevant action to restrict the discharge of pollutants by enterprises and demanded that they raise their wastewater treatment standards.

5.2 Basic components of GMCR

• Different stakeholders and their options in conflicts. There are three DMs: chemical enterprises (CE), ecological environment bureaus (EEB), and local governments (LG). Generally speaking, LG play the role of third-party mediators in conflicts, mainly through administrative mediation, government–court collaboration, and multi-party collaborative mechanisms. Their core is maintaining neutrality, guiding in accordance with the law, and balancing interests, thereby resolving conflicts in a substantive manner. Each DM in conflicts possesses interests, goals, and wishes. In simple terms, the goal of EEB is to ensure that CE increase investment in environmental protection while ensuring normal production by implementing strict environmental protection policies and plans, thereby effectively protecting the fragile local ecology. The interest goal of CE is to maximize personal economic benefits, expand production capacity, and increase profits. Thus, when facing strict environmental policies, CE choose to appeal to the courts. At the same time, LG aim to not only ensure the steady improvement of local economic development but also protect the health of local residents by issuing environmental protection policies (Table 2).
• Feasible states. As shown in Table 2, there are seven different options and three distinct DMs in conflicts, and thereby there are 2⁷ = 128 states based on mathematical calculation. However, the infeasible state refers to the inability to satisfy all necessary constraints in the structure of GMCR, resulting in the inability to become an effective solution or executable operation. Various reasons include insufficient resources, conflicting constraints, violation of target requirements, illegal operations, and initial conditions not being met. Thus, the feasible states of conflict are shown in Table 3.
• Directed graph. This is shown in Figure 5, which describes the movement situation between any two vertices in detail.

TABLE 2. Options of each DM in water resource conflicts in the Yellow River Basin.

TABLE 3. Feasible states in water resource conflicts in the Yellow River Basin.

As described in Table 3, “Y” represents that the corresponding option is selected by the DM; “N” represents the opposite result. “−” means that the state is the same result regardless of whether “Y” or “N” is selected.

As shown in Figure 5, the directed graph of the water resource conflicts in the Yellow River Basin represents the movement between any states for each DM in conflicts. The nine vertices in the directed graph represent the nine different feasible states in Table 3. Moreover, when the choices of a DM who makes up the state are inconsistent, the two generated states in above situation can be moved for that corresponding DM. In other words, the directed edges of a directed graph represent the movement of a given DM between any two states. Directed arcs are used to represent movement between two vertices for three DMs. Among these, some arcs are one-way and some are two-way. A one-way directed arc represents that this movement is irreversible; a two-way directed arc means that this movement is reversible.. In conclusion, directed graphs visually present the logical dependencies and conflicting focal states between conflicting elements using arrow symbols. A one-way arrow can clearly indicate the conditions that must be met to achieve the goal, while a two-way line arrow directly reveals the exclusivity of opposing behaviors. This visualization mechanism simplifies the abstract expression of complex conflicts, making it easier to quickly identify critical paths and conflict points.

5.3 Preference ranking over all states for each DM in conflicts

According to the hydrological statistical report issued by the Yellow River Basin Ecological Environment Administration of the Ministry of Ecology and Environment, we can obtain the current situation and data for water resource conflicts in the Yellow River Basin (https://huanghejg.mee.gov.cn/). At the same time, through on-site visits and research, DM preference data over different states can also be obtained.

In this study, HMBWM is used to rank states for each DM in GMCR. EEB are mainly responsible for formulating, implementing, and supervising environmental policies to ensure the effective implementation of various environmental protection measures. They monitor and evaluate environmental factors such as air, water quality, and soil for the timely tracking of environmental changes, promote the treatment and control of pollution sources, and reduce the discharge of pollutants such as industrial waste gas, sewage, and solid waste. In addition, EEB also bear the responsibility of enforcing and punishing environmental regulations, ensuring that enterprises and the public comply with environmental regulations. At the same time, EEB enhance the environmental awareness of all sectors of society through public education and publicity and promote active public participation in environmental protection actions. By promoting green development, ecological protection, and environmental technology innovation, EEB aim to protect the ecological environment. It is obvious that they are simultaneously stakeholders and managers. Thus, the preferences of EEB are clear and are equal to s ₇ ≻ s ₃ ≻ s ₄ ≻ s ₈ ≻ s ₅ ≻ s ₁ ≻ s ₂ ≻ s ₆ ≻ s ₉ by the direct comparison method.

Next, DMs CE and LG are not professional wastewater treatment personnel and lack professional skills, which also leads to their preference for uncertainty. Faced with this situation, we use the three different solving models of HMBWM to determine the preference ranking for DMs CE and LG.

• Step 1: Nine states {s ₁, s ₂, s ₃, s ₄, s ₅, s ₆, s ₇, s ₈, s ₉} are determined for DM CE.
• Step 2: According to the investigation, the interest goal of CE is to maximize personal economic benefits, expand production capacity, and increase profits. Thus, CE consider s ₆ as the most preferred and s ₉ as the least preferred state.
• Step 3: The HMBO vector is used to represent the preference relation among preferred states s ₆ and other remaining states, which is determined using HMEs.

HMBO = 3 , 4 , 4 , 5 , 2 , 3 , 4 , 1 , 5 , 6 , { 7 , 8 .

• Step 4: The HMOW vector is used to represent the preference relation among least preferred states s ₉ and other remaining states, which is determined by using HMEs.

HMOW = 5 , 6 , 6 , 4 , 6 , 3 , 4 , 7 , 8 , 2 , 3 , { 1 T .

• Step 5: We calculate the weight of feasible states of DM CE.

Firstly, we calculate the geometric mean of HME using Equation 6:

g HMBO = 3.4641 , 4 , 5 , 2 , 3.4641 , 1 , 5 , 6 , 7.4833 ,

g HMOW = 5.4772 , 6 , 4 , 6 , 3.4641 , 7.4833 , 2 , 3 , 1 ,

min ξ g s . t . w B w 1 − 3.4641 ≤ ξ g w B w 2 − 4 ≤ ξ g w B w 3 − 5 ≤ ξ g w B w 4 − 2 ≤ ξ g w B w 5 − 3.4641 ≤ ξ g w B w 7 − 5 ≤ ξ g w B w 8 − 6 ≤ ξ g w B w 9 − 7.4833 ≤ ξ g w 1 w W − 5.4772 ≤ ξ g w 2 w W − 6 ≤ ξ g w 3 w W − 4 ≤ ξ g w 4 w W − 6 ≤ ξ g w 5 w W − 3.4641 ≤ ξ g w 6 w W − 7.4833 ≤ ξ g w 7 w W − 2 ≤ ξ g w 8 w W − 3 ≤ ξ g ∑ j = 1 9 w j = 1 w j ≥ 0 , for all j .

We obtain w=(0.1390, 0.1225, 0.0740, 0.1673, 0.0899, 0.2702, 0.0563, 0.0516, 0.0292). Thus, the preference ranking for DM CE is s ₉≺s ₈≺s ₇≺s ₃≺s ₅≺s ₂≺s ₁≺s ₄≺s ₆.

Finally, the HMBWM method also can be used to obtain the preference ranking for DM LG.

HMBO = 6 , 3 , 5 , 2 , 4 , 5 , 6 , 5 , 1 , { 7 , 8 ,

HMOW = 2 , 3 , 3 , 2 , 6 , 5 , 2 , 6 , 7 , 8 , { 1 T .

Based on this solving in Model 10, we have w=(0.0543, 0.0984, 0.0643, 0.1700, 0.1183, 0.0644, 0.0965, 0.3030, 0.0307). Thus, as for DM LG, we have s ₉≺s ₁≺s ₃≺s ₆≺s ₇≺s ₂≺s ₅≺s ₄≺s ₈.

5.4 Stability analysis of water resource conflicts in the Yellow River Basin

Four elements are determined in the above content, which establish the prerequisites for stability analysis. Thus, based on research results in Section 5.3 and Definitions 18−22, the hesitant multiplicative stability analysis results of water resource conflicts are shown in Table 4.

TABLE 4. Hesitant multiplicative stability analysis results in water resource conflicts in the Yellow River Basin.

As shown in Table 3, DMs 1–3, respectively, represent DM EEB, DM CE, and DM LG. Note that HME represents that the state is hesitant multiplicative stable for each DM. State s ₄ is only hesitant multiplicative stable for resolving real-life water resource conflicts in the Yellow River Basin. Specifically, DM EEB selects to revise the existing strict environmental protection policies and plans, which can help CE and LG repair environments polluted by chemical industry wastewater. DM CE select to accept existing strict environmental protection policies and plans, which can simultaneously ensure economic development and environmental protection. DM LG hold a consultative attitude to EEB. Finally, DMs in conflict make concessions to each other to achieve reconciliation, thereby promoting a win–win situation.

A stable state provides an efficient and intuitive framework for identifying equilibrium solutions in complex conflicts by replacing numerical calculations with logical verification. The core value of a stable state lies in transforming abstract policy interactions into operational stability verification processes, ultimately guiding DMs to escape from local optimal traps and achieve system-level conflict resolution. Furthermore, the stable state provides DMs with a systematic tool from individual strategy selection to global conflict management through behavior rationalization guidance, preference structure optimization, and dynamic evolution prediction. Its core value lies in transforming abstract conflict problems into operable stability verification frameworks, thus ultimately achieving balanced decision-making with controllable risks.

Ningxia government in Yellow River Basin should adhere to innovation-driven development and achieve innovative development. LG should focus on scientific experiments and technological breakthroughs in areas such as water safety, ecological protection, and water and sediment regulation. LG in Yellow River Basin should strengthen regional division of labor and cooperation to achieve coordinated development. It should closely combine its own comparative advantages and development stages to determine its development strategy, optimize and adjust the regional economic and productivity layout on the premise of ecological protection, and promote reasonable division of labor in upstream, midstream, and downstream regions. LG should adhere to ecological priority and achieve green development. It should regard water resources as the maximum rigid constraint and build an intensive and economic development path that effectively guarantees water security, efficiently utilizes water resources, and significantly improves water ecology.

6 Comparative analysis between GMCR with HMPRs and other existing GMCR

The stability condition of HMNash stability is HM R k + s = ∅. It is evident that the complexity of HMNash stability is O(m). Next, the complexity of HMGMR are determined as follows: (1) UI list satisfies |H M R k + s ≤m−1|; (2) x ₁∈ H M R k + s, N − k needs at most (n−1)∙O (m ^*L*−1) to determine R N − k s 1; (3) the comparison between s and s ₂ for DM k is at most m−1, where s 2 ∈ R N − k s 1. The complexity of HMGMR stable is m+(m−1)[(n−1)∙O (m ^*L*−1)+m−1]=(n−1)∙O(m ^*L*). Finally, the complexities of HMSMR, HMSEQ, and HMSSEQ can be calculated same way (Table 5).

TABLE 5. Computational complexities of the five hesitant multiplicative stability definitions.

As shown in Table 4, new proposed GMCR with HMPRs significantly reduce the computational complexity of real-life conflicts through strategy dimensionality reduction, preference simplification, and graph structure optimization. The core value of new proposed GMCR lies in transforming high-dimensional dynamic games into operable stability verification processes, thus balancing computational efficiency and decision accuracy. Particularly when using graph models with lower computational complexity to solve real-world conflicts, the conflict analysis graph model will generate executable equilibrium solutions based on a stability verification framework, avoiding the dilemma of conflict resolution solutions becoming “theoretically optimal but practically infeasible.”

GMCR with HMPRs, original GMCR (Kilgour et al., 1987), fuzzy GMCR (Bashar et al., 2012), and GMCR with TFNs (Sun and Wang, 2025) are selected to carry out comparative analysis (Table 6). Note that original GMCR is the basic GMCR with crisp preference. Fuzzy GMCR and GMCR with TFNs are, respectively, the classic graph model with uncertain preferences and the latest graph model with uncertain preferences.

TABLE 6. Comparison among three different GMCR models.

As shown in Table 6, compared with the existing three different types of graph models, the GMCR with HMPRs proposed here has multiple advantages, including ensuring the completeness of conflict analysis results, analyzing uncertain preferences, proposing new preference acquisition methods, and characterizing DM hesitancy, which makes its applicability wider and more general.

7 Conclusions and future research

First, considering that existing preference acquisition skills in traditional GMCR cannot be applied to uncertain environments, this study initially designs HMBWM to obtain preference ranking. Second, considering the uncertainty and complexity of DM preference cognition and the real-life background of conflict problems, HMPR form is chosen to characterize DMs’ uncertain evaluation over states, which can more accurately reflect DMs’ real life preference tendency. Third, within the GMCR with HMPRs, five hesitant multiplicative stability definitions have been proposed to solve real world conflicts in uncertain complex environments. Finally, water resource conflicts in the Yellow River Basin were applied to fully illustrate the effectiveness and practicality of both GMCR with HMPRs and HMBWM. However, there are various limitations in this paper, as follow.

• In the context of water resource conflicts in the Yellow River Basin, each DM may be considered as CDMs, DM CE may be collections of enterprises with different production properties, and the most preferred state and the least preferred state for DM CE may be different.
• In HMBWM, the evaluation values for the preferences between two different states are subjectively given, lacking theoretical guidance.
• The logical representation of stability definitions in GMCR with HMPRs has high computational complexity and is difficult to program and apply to real-world conflict problems.

Finally, there are still some research shortcomings in this study, and future research can be conducted from the following perspectives.

• DMs may be composed of different experts, known as CDMs. It is feasible to consider the internal consensus of CDMs in order to reach a consensus on effective conflict resolution.
• Third-party regulation and evaluation have significant contributions to conflict resolution. How to construct a reasonable third-party mediation model is of decisive significance for conflict resolution, and the model construction process needs to consider the uncertainty of DMs. The inverse problem analysis framework of GMCR provides a supporting framework for third-party mediation.
• In order to reduce computational complexity and facilitate programming, logical expressions of hesitant multiplicative stability should be designed in future research and applied to solve real-world conflicts.

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Glossary

BO Best-to-others BWM Best–worst method CE Chemical enterprises DM Decision maker EEB Ecological environment bureaus GMCR Graph model for conflict resolution GMR General metarationality HFSs Hesitant fuzzy sets HMBWM Hesitant multiplicative best–worst method HMBO Hesitant multiplicative best-to-others HMOW Hesitant multiplicative others-to-worst HMPRs Hesitant multiplicative preference relations HME Hesitant multiplicative element HMGMR Hesitant multiplicative general metarational HMNash Hesitant multiplicative Nash HMS Hesitant multiplicative set HMSEQ Hesitant multiplicative sequential stability HMSMR Hesitant multiplicative symmetric metarational HMSSEQ Hesitant multiplicative symmetric sequential stability HMUIL Hesitant multiplicative unilateral improvement list HMURL Hesitant multiplicative unilateral reachable list iff If and only if LG Local governments Nash, R Nash OW Others-to-worst SEQ Sequential stability SMR Symmetric metarationality SSEQ Symmetric sequential stability

Footnotes

Edited by: Enoch Akwasi Kosoe, SD Dombo University of Business and Integrated Development Studies, Ghana

Reviewed by: Junren Luo, National University of Defense Technology, China

Sivajothi Ramalingam, Saveetha Engineering College, India