Conflict analysis of disputes in livelihood vulnerability assessment of flood using fuzzy TOPSIS method and GMCR with triangular fuzzy numbers

Abstract

Flooding is a common natural disaster that poses a direct threat to the livelihoods of residents in developing countries who lack sufficient funding and technology. In the system for assessing the livelihood vulnerability (LHV) of homeless and impoverished populations, decision-makers (DMs) come from various departments. Additionally, DMs may involve one or more stakeholders. Thus, disputes in the negotiation process of LHV assessment have become very frequent, which will seriously prevent residents from restoring their production and life. In addition, the complex natural attributes of flood disasters and the inherent ambiguity of human choice exacerbate the inhomogeneity and uncertainty of disputes in LHV assessment. The traditional graph model for conflict resolution (GMCR) was proposed to resolve real-life conflicts. To alleviate the livelihood damage caused by floods to residents in complex uncertain environments, the novel GMCR with TFNs is constructed. Specifically, first, triangular fuzzy numbers (TFNs) are used to character DM’s preference on different states in LHV assessment. Then, the fuzzy Technique for Order Preference by Similarity to Ideal Solution (F-TOPSIS) method is constructed to rank all states in uncertain environments. Next, a set of stability concepts is determined in GMCR with TFNs to obtain equilibrium of real-life disputes. Finally, the proposed method is applied in real-life disputes in livelihood vulnerability assessment of flood in the Yangtze River basin in China. The research not only promotes the theory development of GMCR but also provides a theoretical reference for developing countries to solve flood crises and enhance the livelihood security of residents.

Introduction

Climate change has gradually become a key scientific issue affecting human survival and development¹. The complexity and uncertainty of climate parameter variations make natural environment and ecosystems more fragile and difficult to analyze. Climate is one of the important factors affecting people’s lives and production, as well as, production and lives way constitute the main manifestation of livelihood. The term ‘‘livelihood’’ refers to the skills, abilities, and goals that stakeholders must possess and master to pursue a certain way of life. It is the most fundamental influencing factor that determines a person’s quality of life and life goals². Furthermore, the most critical influencing factor for evaluating livelihoods is vulnerability. The term “vulnerability” mainly emphasizes the resilience, coping, and recovery capabilities of human socio-economic systems when affected by disasters, and is determined by a set of factors, such as cultural, political, and psychological factors. It is evident that it is a multidimensional factor influencing indicator that is difficult to model and analyze³. Particularly, the intensification of environmental pollution has exacerbated the hazards of natural disasters, while pandemics such as COVID-19 and inadequate healthcare infrastructure have also increased vulnerability⁴. In general, climate anomalies lead to changes in atmospheric circulation, resulting in increased precipitation and flooding in certain areas. The main hazards of floods include casualties, property damage, crop damage, water pollution, disease transmission, and social issues. The hazards of floods undoubtedly affect the livelihoods of local people. Thus, it is crucial and meaningful to research the impact of floods on residents’ livelihoods from the perspective of vulnerability to guide each stakeholder to enhance recovery capabilities and minimize losses to the greatest extent possible.

Generally, livelihood vulnerability (LHV) represents the level to which a person’s lifestyles are adversely affected by both physical and social factors⁵. To be specific, physiological vulnerability refers to the degree to which factors such as rising temperatures, environmental pollution, and glaciers melting affect the human environment⁶. Social vulnerability refers to the impact of social phenomena such as social integration and human interaction on the ability of individuals or communities to recover from natural disasters, thereby affecting their vulnerability⁷. Note that this paper focuses on LHV of flood, and determines and analyzes its physical and social factors. From the perspective of climate, the causes of floods are usually due to high-intensity, prolonged rainfall and processes such as melting snow and ice caused by rising temperatures. Extreme precipitation is often related to weather systems such as abnormal atmospheric circulation, typhoons, and atmospheric river water vapor transport⁸. Unreasonable planning and management of land use are also important reasons for the occurrence of floods and waterlogging disasters. In the process of urbanization, a large number of non-permeable areas (such as buildings, roads, etc.) will reduce the opportunities for natural infiltration, leading to rainwater gathering in low-lying areas. In addition, excessive land development, deforestation, and ecosystem destruction can weaken the natural conservation function of land and increase the risk of floods and waterlogging disasters⁹.

One of the most important works in LHV assessment is to analyze the impact of natural disasters such as floods on the livelihoods of different stakeholders and explore their livelihood changes, vulnerability levels, and regulatory strategies¹⁰. Furthermore, in the management system of evaluating LHV in countries or regions affected by flood disasters, stakeholders may involve different social classes and work departments. In general, different decision-makers (DMs) with different cultural and knowledge backgrounds are prone to conflict in the process for identifying relevant criteria for LHV assessment, which will seriously hinder the effective implementation of flood control and reduce their livelihood vulnerability. Thus, negotiation among DMs is useful skills for calculating solutions that is preferred and accepted by all stakeholders^11,12. Based on the above described background, disputes in livelihood vulnerability assessment of flood can be seen as highly structured conflict problems. That is to say, various experts actively participate into the decision-making process, show their preference attitudes over states, and then determine an equilibrium state of conflicts by using stability analysis based on a set of stability concepts^{13,14,15,16,17}.

Just as this article researches disputes in livelihood vulnerability assessment of flood, conflicts mean that when two or more DMs possess different attitudes on negotiation to maximize their interests, which produces contradiction¹⁸. Clearly, conflicts are a common natural phenomenon, ranging from arguments between parents and children in families to military wars between nations. The significance of resolving conflicts lies in promoting social development and progress, meeting the growing needs of the people for a better life, and promoting social fairness, justice, and comprehensive human development. Thus, the resolution methods for conflict problems have always been a focus of attention for researchers in the field of decision analysis. Various conflict analysis methods, including metagame theory¹⁹ and F-H conflict analysis²⁰, drama theory^21,22, and graph model for conflict resolution (GMCR)²³, have been proposed from classical game theory²⁴ to resolve disputes. Among them, GMCR is considered the latest conflict analysis method, which has the characteristic of simple operation and wide application range. GMCR overcomes the limitation of existing conflict analysis methods that require much precise data by using less relative preference^25,26,27,28. When GMCR is used to investigate conflicts, there exist two steps: first is the modeling stage, and then is the analysis stage. Then, Nash stability (Nash, R)^29,30, general metarationality (GMR)¹⁹, symmetric metarationality (SMR)¹⁹, and sequential stability (SEQ)²⁰ are proposed in traditional GMCR paradigm to describe DM’s behavior patterns³¹. As for the specific research object, after the floods in the river basin in a certain region, the assessment methods established using precise digital data faced significant challenges in assessing the livelihood vulnerability of homeless or poor persons. Thus, this article uses GMCR to carry out stability analysis in disputes in livelihood vulnerability assessment of flood from the perspectives of qualitative analysis and obtain a set of feasible policy suggestions.

Within traditional GMCR, the representation form of DM’s preference ranking is mainly divided into two categories, such as cardinal payoffs and ordinal preferences³². In fact, DMs’ cardinal utilities are hard to determine. Fortunately, conflict analysis can also be conducted with only ordinal rankings. Thus, three preference ranking methods, including direct ranking, option weight, and option prioritization, have been constructed to calculate the ordinal preference in GMCR. However, with the increasing frequency of human interaction and the complexity of conflict issues, these existing preference ranking methods are difficult to apply. To overcome the above shortcomings, various theories have been embedded into the framework of GMCR to obtain DM’s real life preference relations^33,34,35. However, the existing researches have not taken into account two aspects: The first situation is not taking into account the uncertain preferences of DMs. The second situation does not take into account the composite DMs situation formed by various DMs with common interests in the conflict^18,36. To overcome these problems, in this paper, fuzzy Technique for Order Preference by Similarity to Ideal Solution (F-TOPSIS) is first constructed into the GMCR paradigm to obtain DM’s preference ranking in complex uncertainty environment. Recall that the existing research on using traditional TOPSIS to carry out assessment rarely considers the uncertainty environment³⁷. In a word, F-TOPSIS proposed not only overcomes difficulties of traditional TOPSIS method but also is applied to the selection problem in different scenarios^38,39. Note that multi-objective decision method has received a lot of attention and has a wide range of applications^40,41.

It is evident that DM’s preference relation in traditional GMCR is a crisp preference. However, traditional GMCR cannot accurately characterize DM’s uncertainty and cannot used to obtain solutions of conflicts in complex and unknown environments. To promote the further development of GMCR, Li, et al.⁴² constructed a set of novel stability concepts based on new proposed uncertain preferences. More importantly, fuzzy preference, as the most classic and commonly used uncertain preference, has also been incorporated into GMCR to describe uncertainty⁴³. Subsequently, various uncertain preferences were incorporated into GMCR by multiple excellent scholars in different ways and applied to conflicts in different fields^36,44,45,46. Recently, triangular fuzzy numbers (TFNs) have attracted attention due to their ability to effectively handle problems in uncertain environments^47,48. However, DMs’ preferences in GMCR have not yet been represented by using TFNs. Thus, in this paper, to more accurately characterize DM’s uncertainty in conflicts, TFNs are embedded into GMCR and then propose a set of stability definitions with TFNs, including triangular fuzzy Nash (TFNash, TFR), triangular fuzzy GMR (TFGMR), triangular fuzzy SMR (TFSMR), and triangular fuzzy SEQ (TFSEQ). The Yangtze River basin is an area of concentrated agriculture, industry, and commercial trade in China. Floods in the Yangtze River basin will cause more serious casualties, property damage, crop damage, and water pollution problems, which will bring huge losses to the lives and health of local residents and development of economic society^49,50. Based on this, the disputes in livelihood vulnerability assessment of flood in Yangtze River basin of China are selected as research objects, and the stability analysis results not only prove the feasibility of combining F-TOPSIS and GMCR with TFNs but also provide effective policy recommendations for reducing secondary disasters caused by floods and improving the livelihood level of the people. The important contributions of the article are listed below:

(1) This paper first a new preference ranking method named F-TOPSIS and embed this approach into GMCR structure, which can promote and develop the theoretical contribution of GMCR in the complex fuzzy environment.

(2) TFNs is first considered into GMCR to character DM’s uncertain preferences, which provides a new skill to describe DM’s interaction behaviors. Four different triangular fuzzy stability definitions are constructed to resolve conflicts in uncertain environments.

(3) This paper first researches disputes in livelihood vulnerability assessment of flood from the perspective of quantitative analysis by using GMCR method, which is different from traditional research ideas. The calculation process and analysis results become an example for investigating similar conflicts in different fields.

The structure of the remaining articles is arranged as follows: Sect. 2 reviews the basic concepts of linguistic variables, fuzzy sets, fuzzy numbers, and GMCR. In Sect. 3, F-TOPSIS is first constructed into GMCR to rank all states. Section 4 defines a set of stability definitions called triangular fuzzy stability in GMCR. In Sect. 5, the proposed method is applied in disputes in livelihood vulnerability assessment of flood in Yangtze River basin of China. Finally, some conclusions and future research directions are given in Sect. 6.

Preliminaries

Linguistic variable and TFNs

Linguistic variable in group decision-making refers to that preference values are represented by natural or artificial language words. It allows us to show DM’s preference relations by using a set of human-friendly languages such as “low”, ‘‘medium’’, or ‘‘high’’. For example, linguistic terms including cold, cool, warm, and hot, are used to show personal DM’s feelings on “temperature”.

To character and represent the inherent internal uncertainty in linguistic terms, linguistic variables are also represented by using fuzzy sets. Fuzzy sets provide a new skill to represent uncertainty by using degrees of membership. Furthermore, TFNs, as an important subclass of fuzzy numbers, used function composed by lower bound, a peak, and an upper bound to show DM’s uncertainty.

Definition 1 ⁵¹

The fuzzy set X on set A is described as X={(x, f_X(x)):x∈A$\}$, where f_X:A→[0, 1] is the membership function of the fuzzy set X.

Definition 2 ⁵²

The TFNs X can be described by a set of triplet (α, β, γ), and it has a membership as shown in the following.

$${f_X}(x)=\left\{ \begin{gathered} 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x<\alpha \hfill \\ \frac{{x - \alpha }}{{\beta - a}},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha \leqslant x \leqslant \beta {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \hfill \\ \frac{{\gamma - x}}{{\gamma - \beta }},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \beta \leqslant x \leqslant \gamma \hfill \\ 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x>\gamma \hfill \\ \end{gathered} \right.$$

where α, β, γ  are real number with the nature of α<β<γ.

Furthermore, there exist two different TFNs, including X₁=(α₁, β₁, γ₁) and X₂=(α₂, β₂, γ₂). A set of operational laws between TFNs is shown as follows:

$$\begin{aligned}{X_{\text{1}}}+{X_{\text{2}}}&=({\alpha _{\text{1}}}+{\alpha _{\text{2}}},{\beta _{\text{1}}}+{\beta _{\text{2}}},{\gamma _{\text{1}}}+{\gamma _{\text{2}}})\\{X_{\text{1}}} - {X_{\text{2}}}&=({\alpha _{\text{1}}} - {\alpha _{\text{2}}},{\beta _{\text{1}}} - {\beta _{\text{2}}},{\gamma _{\text{1}}} - {\gamma _{\text{2}}}\\{X_{\text{1}}} \times {X_{\text{2}}}&=({\alpha _{\text{1}}}{\alpha _{\text{2}}},{\beta _{\text{1}}}{\beta _{\text{2}}},{\gamma _{\text{1}}}{\gamma _{\text{2}}})\\{X_{\text{1}}}/{X_{\text{2}}}&=({\alpha _{\text{1}}}/{\beta _{\text{2}}},{\beta _{\text{1}}}/{\beta _{\text{2}}},{\gamma _{\text{1}}}/{\alpha _{\text{2}}})\\k \times {X_{\text{1}}}&=(k{\alpha _{\text{1}}},k{\beta _{\text{1}}},k{\gamma _{\text{1}}})\end{aligned}$$

In addition, the distance between two TFNs is shown as follows:

$$D({X_{\text{1}}},{X_{\text{2}}})=\sqrt {\tfrac{1}{3}[{{({\alpha _1} - {\alpha _2})}^2}+{{({\beta _1} - {\beta _2})}^2}+{{({\gamma _1} - {\gamma _2})}^2}]}$$

Finally, the conversion between linguistic variable and TFN is shown in Table 1².

Table 1 Conversion between linguistic variable and TFN.

GMCR

In order to resolve conflicts and provide suggestion for each stakeholder, GMCR is proposed and represented as $\{ N,S,D,P\}$, where $N=\{ 1,2, \ldots ,m\}$ is composed of a set of stakeholders or experts or DMs, $S=\{ {s_1},{s_2}, \ldots ,{s_n}\}$ is a set of feasible states, D=(S, A_k) represents the directed graph and ${A_k} \subseteq S \times S$ is the directed arc, and P is preferences, specifically, s_i≻_ks_j indicates that state s_i is preferred than state s_j for DM k and s_i~_ks_j implies that DM k is indifferent between state s_i and state s_j.

After determining four basic elements of GMCR, the stability analysis stage is to determinate whether a given feasible state is acceptable for DM by using limited step movement according to any type of stability analysis definition. The application step of GMCR is shown in Fig. 1.

Fig. 1

The modeling and analysis steps of the GMCR.

Next, some definitions have been described to review basic stability concepts.

Definition 3

In GMCR, the unilateral reachable list is defined as ${R_k}(s)=\{ {s_1} \in S,(s,{s_1}) \in {A_k}\}$. Any movement in R_k(s) belongs to unilateral move.

Definition 4

In GMCR, the unilateral improved reachable set is defined as $R_{k}^{+}(s)=\{ {s_1} \in {R_k}(s),{s_1}{ \succ _k}s\}$. Any movement in $R_{k}^{+}(s)$ belongs to unilateral improvement move.

Let a coalition H⊆N, satisfies$\left| H \right| \geqslant 2$. Then, R_H(s) and $\:{R}_{H}^{+}$(s) are defined by using an iterative method. Note that all DMs that are played in the legal sequence of R_H(s) are represented as Ω_H(s, s_i). By using same way, we can obtain $\Omega _{H}^{+}(s,{s_i})$.

Definition 5

Suppose that a coalition H⊆N and a state s∈S. Thus, R_H(s) is determined as follows.

(1) If $k \in H$ and ${s_1} \in {R_k}(s)$, then ${s_1} \in {R_H}(s)$ and $k \in {\Omega _H}(s,{s_1})$.

(2) If $k \in H$, ${s_1} \in {R_H}(s)$, and ${s_2} \in {R_k}({s_1})$, then we obtain ${\Omega _H}(s,{s_1}) \ne \{ k\}$,${s_2} \in {R_H}(s)$ and $k \in {\Omega _H}(s,{s_2})$.

The difference between the R_H(s) and $\:{R}_{H}^{+}$(s) is that the state reached in $\:{R}_{H}^{+}$(s) is better than the initial state.

Definition 6

A state s is Nash stability for DM k if and only if (iff) there exist $R_{k}^{+}(s)=\emptyset$, represented as $s \in S_{k}^{{Nash}}$.

Definition 7

A state s is GMR for DM k iff for each state ${s_1} \in R_{k}^{+}(s)$ there has at least one state ${s_2} \in {R_{N - \{ k\} }}({s_1})$ satisfies ${s_2}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \prec } s$, represented as $s \in S_{k}^{{GMR}}$.

Definition 8

A state s is SMR for DM k iff for each state ${s_1} \in R_{k}^{+}(s)$ there has at least one state ${s_2} \in {R_{N - \{ k\} }}({s_1})$ satisfies ${s_2}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \prec } s$ and ${s_3}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \prec } s$ for all state ${s_3} \in {R_k}({s_2})$, represented as $s \in S_{k}^{{SMR}}$.

Definition 9

A state s is SEQ for DM k iff for each state ${s_1} \in R_{k}^{+}(s)$ there has at least one state ${s_2} \in R_{{N - \{ k\} }}^{+}({s_1})$ satisfies ${s_2}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \prec } s$, represented as $s \in S_{k}^{{SEQ}}$.

Definitions 6− 9 are constructed according to personal DM’s choice and attitudes, and need to fully consider the DM’s reactions when they are involved in potential risks. Among them, Nash stability means that the focus DM does not care about risks and would choose any possible actions to obtain interests. The remaining three basic stability definitions describe DMs’ action with foresight.

Remark 1

When the conflict problems just include two different DMs, Definitions 6 − 9 are transformed to describe DM’s interaction in conflicts with two DMs.

Constructing a new preference elicitation technology in GMCR for resolving disputes in livelihood vulnerability assessment of flood

We first briefly review existing preference ranking method. Then, a new preference elicitation technology called F-TOPSIS is integrated into GMCR to obtain preference ranking in complex and uncertainty environments.

Review of three preference ranking methods in traditional GMCR

(1) Option weight: A set of option $O=\{ {o_1},{o_2}, \ldots ,{o_t}\}$ is proposed to describe different actions in conflicts. For a DM k, each option o_i is assigned a fixed numerical weight W(o_i) based on its personal importance to that DM. The more preferred the option is, the greater the weight it gives. In option weighting, a calculation formulation M(s) for each state s is determined based on the known option weight. The final scores of each state are used to rank feasible states in conflict problems. Then, we have

$$M(s)=\sum\limits_{{{o_i} \in O}} {W({o_i}) \cdot s({o_i})} ,\;~\forall s \in S{\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 1 \leqslant i \leqslant t$$

(1)

After using Eq. (1) to score each state in the conflicts, we will sort all states by using the quick-sort method from the most preferred to least preferred.

(2) Option prioritization: In the solving step of priority prioritization, the priority ordered set of preference statements is firstly constructed to describe feasible states for each DM. It is evident that the preference statement of each state has a unique truth value, i.e. True (T) or False (F). Then, suppose that a set of statements ${\Omega _1},{\Omega _2}, \ldots ,{\Omega _r}$ is used to determine the preferences. Given two different feasible states s₁, s₂ in conflict problems, the state s₁ is considered superior to the state s₂ iff there exists the following condition. Mathematically,

$$\begin{aligned}{\Omega _1}({s_1})&={\Omega _1}({s_2})\\{\Omega _2}({s_1})&={\Omega _2}({s_2})\\& \vdots\\ {\Omega _{j - 1}}({s_1})&={\Omega _{j - 1}}({s_2})\\{\Omega _j}({s_1})&=T\;{\text{and}}\;{\Omega _j}({s_2})=F\end{aligned}$$

where $0<j \leqslant r$. On the one hand, when the positions of T and F in the above last constraint condition are replaced, it indicates that s₂ is preferred to s₁. On the other hand, if there is no exist j in the last constraint condition and then two states s₁ and s₂ have equal preference.

The preference statements can be divided into three categories, including nonconditional, conditional, and bi-conditional. Generally speaking, a nonconditional statement is composed of the different selected options. The connectives, such as negation (“not” or −), conjunction (“and” or &), and disjunction(“or” or |), are used to join different options in the nonconditional statement. Brackets (“(” and “)”) are used to prioritize different operations in a statement. In addition, the conditional and bi-conditional statements are expressed as a combination of two different nonconditional statements connected of “IF” or “IFF”. Its truth depends on the truth value of two different composition factors based on truth tables, which can be used to describe the option prioritization based on logical connectives. A score function $\Psi (s)$ is proposed to rank states according to the truth values of statements in a given state. Let p be the maximum value of the statement and ${\Psi _j}(s)$ is the incremental score function for state s according to the given statement ${\Omega _j}$. Then,

$${\Psi _j}(s)=\left\{ \begin{gathered} {2^{p - j}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{if}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\Omega _j}(s)=T \hfill \\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{otherwise}} \hfill \\ \end{gathered} \right.$$

(2)

and

$$\Psi (s)=\sum\limits_{{j=1}}^{k} {{\Psi _j}(s)}$$

(3)

where $0<j \leqslant p$. Finally, compare the score of different states, and all states can be sorted from the best to the worst using the quick sort method.

(3) Direct ranking: When using GMCR to analyze the conflict problems, direct ranking method allows DMs to directly determine preference rankings based on personal preferences.

Constructing F-TOPSIS to determine DM’s preference ranking in GMCR

TOPSIS is a prominent distance-based MCDM technique. Nowadays, TOPSIS has been connected with fuzzy theory to deal with decision-making with uncertainty. Particularly, it is evident that TFN is a very rational skill for describing DM’s inherent ambiguity and uncertainty in decision-making process. Thus, when all ratings and weights are determined by TFNs, we design and introduce solving steps of F-TOPSIS technique.

Step 1: Determining each DM’s decision matrix.

Suppose that there exist k experts (E₁, E₂, …, E_k) in a composite DM, m alternatives (A₁, A₂, …, A_m), and n criteria (c₁, c₂, ..., c_n) in conflicts. Based on this, the decision matrix is shown as follows:

$$X={\left( {\begin{array}{*{20}{c}} {{x_{11}}}&{{x_{12}}}& \cdots &{{x_{1n}}} \\ {{x_{21}}}&{{x_{22}}}& \cdots &{{x_{2n}}} \\ \vdots & \vdots & \ddots & \vdots \\ {{x_{m1}}}&{{x_{m2}}}& \cdots &{{x_{mn}}} \end{array}} \right)_{m \times n}}$$

where x_ij in matrix X indicates that TFN characterized for the alternative A_i with respect to c_j criterion.

Step 2: Aggregating the DMs’ evaluations.

Suppose that there exits k different experts, the criterion c_j has been evaluated, represented as w_jt=(α_jt), where t = 1, 2, ..., k. Furthermore, the aggregated fuzzy weight is represented as W_j=$(a_{j}^{*},\beta _{j}^{*},\gamma _{j}^{*})$, there exists

$$\alpha _{j}^{*}=\mathop {\hbox{min} }\limits_{k} \{ {a_{jt}}\} ,\;\beta _{j}^{*}=\frac{1}{k}\sum\nolimits_{{t=1}}^{k} {{\beta _{jt}}} ,\;\gamma _{j}^{*}=\hbox{max} \{ {\gamma _{jt}}\}$$

(4)

In the same way, we use ${x_{ijt}}=({\alpha _{ijt}},{\beta _{ijt}},{\gamma _{ijt}})$ to represents the fuzzy rating between alternatives A_i and the criteria c_j for expert E_k. Furthermore, the aggregated fuzzy rating is y_ij=$(\alpha _{{ijt}}^{*},\beta _{{ijt}}^{*},\gamma _{{ijt}}^{*})$, there exits

$$\alpha _{{ij}}^{*}=\mathop {\hbox{min} }\limits_{k} \{ {a_{ijt}}\} ,\;\beta _{{ij}}^{*}=\frac{1}{k}\sum\nolimits_{{t=1}}^{k} {{\beta _{ijt}}} ,\;\gamma _{{ij}}^{*}=\hbox{max} \{ {\gamma _{ijt}}\}$$

(5)

Step 3: Normalizing the DM.

We use linear scale transformation to normalize each elements in Y=(y_ij)_m×n. Particular, we suppose that c_j is benefit criteria, and thereby we have

$${\gamma _{ij}}=\left( {\frac{{a_{{ij}}^{*}}}{{{\gamma ^*}}},\frac{{\beta _{{ij}}^{*}}}{{{\gamma ^*}}},\frac{{\gamma _{{ij}}^{*}}}{{{\gamma ^*}}}} \right)$$

(6)

where ${\gamma ^*}={\hbox{max} _k}\{ \gamma _{{ij}}^{*}\}$. By the same way, we suppose that c_j is cost criteria, and thereby we have

$${\gamma _{ij}}=\left( {\frac{{{a^*}}}{{\gamma _{{ij}}^{*}}},\frac{{{a^*}}}{{\beta _{{ij}}^{*}}},\frac{{{a^*}}}{{\alpha _{{ij}}^{*}}}} \right)$$

(7)

where ${\alpha ^*}={\hbox{min} _k}\{ \alpha _{{ij}}^{*}\}$. Based on this, the normalized expert is represented as R=(r_ij)_m×n.

Step 4: Computing weighted normalized DM.

According to R and W_j, we obtain V=(w_ij)_m×n. Thus, we have

$${v_{ij}}={W_j} \times {r_{ij}},\;i\,=\,{\text{1}},{\text{ 2}}, \ldots ,m{\text{ and}}\,j\,=\,{\text{1}},{\text{ 2}}, \ldots ,n$$

(8)

Step 5: Computing the fuzzy positive ideal solution (FPIS) and fuzzy negative ideal solution (FNIS).

Suppose that there exits two equations, as shown in the following

$${A^+}=(v_{1}^{+},v_{2}^{+},\ldots,v_{n}^{+})$$

(9)

and

$${A^ - }=(v_{1}^{ - },v_{2}^{ - },\ldots,v_{n}^{ - })$$

(10)

where $v_{j}^{+}={\hbox{max} _i}\{ {v_{ij}}\}$ and $v_{j}^{ - }={\hbox{min} _i}\{ {v_{ij}}\}$.

Step 6: Calculating the distance.

The distance of each alternatives A_i is determined by FPIS and FNIS, and thereby we have

$$D_{i}^{+}=\sqrt {\sum\nolimits_{{j=1}}^{n} {D({v_{ij}},v_{j}^{+})} } ,\,\;i\,=\,{\text{1}},{\text{ 2}}, \ldots ,m$$

(11)

and

$$D_{i}^{ - }=\sqrt {\sum\nolimits_{{j=1}}^{n} {D({v_{ij}},v_{j}^{ - })} } ,\quad i\,=\,{\text{1}},{\text{ 2}}, \ldots ,m$$

(12)

where D(, ) represents the distance between two TFNs.

Step 7: Calculating the closeness coefficient (CC).

The value of CC of each alternatives A_i is determined as follows:

$$C{C_i}=\frac{{D_{i}^{ - }}}{{D_{i}^{ - }+D_{i}^{+}}},\quad i\,=\,{\text{1}},{\text{ 2}}, \ldots ,m$$

(13)

Step 8: Ranking all alternatives.

The alternatives are ranked according to their CC values in descending order. In other words, the higher the value of CC_i, the higher the corresponding alternatives ranking.

Based on this, as shown in Fig. 2, an F-TOPSIS method is constructed to obtain the preference ranking, which makes the acquisition process of preference ranking more authentic and trustworthy. It is interesting noting that, although the new proposed preference ranking methods in GMCR make the analysis process is considered more complicated than traditional GMCR. It also provides a more objective preference solving method, which ensures the preference ranking result and even conflict resolution more accurately and reasonable.

Fig. 2

Framework of the combining of F-TOPSIS and GMCR method.

The calculation process of F-TOPSIS in resolving disputes in livelihood vulnerability assessment of flood

Considering the characteristics of livelihood vulnerability assessment of flood, F-TOPSIS needs to consider some necessary influencing physical and social factors in the application process to adapt to the complexity and specificity of LHV. To achieve our research aims, the calculation process of F-TOPSIS in resolving disputes in livelihood vulnerability assessment of flood has been divided into three steps: First, a set of appraisal criteria is determined to evaluate LHV; second, a set of linguistic variable is used to weight criteria and evaluate ratings for evaluating LHV; third, the linguistic representation were converted into FTNs, which makes that qualitative information have been transformed into quantitative one. Based on this, the ranking over all feasible states has been determined by using F-TOPSIS method.

If severe flooding occurs in a river basin, we assume that there are three action plans, that are equal to feasible states, represented as s₁, s₂, and s₃, for implementing LHV in that basin. In order to rank three action plans using F-TOPSIS method, we first identified the criteria that affect LHV from the different perspectives, as shown in Table 2.

Table 2 Different criteria are determined in LHV assessment.

Next, 12 experts from different fields were invited to provide their own qualitative analysis of the LHV problem, represented from E₁ to E₁₂. The opinions over three action plans for all experts are represented by using linguistic preferences, as shown in Tables 3 and 4.

Table 3 All experts’ preferences for the criteria.

Table 4 All experts’ ratings for the alternatives.

To explain the aggregation process, we use state s₁ as an example. Three DMs consider that s₁ is very highly preferred, four DMs consider it highly preferred, three DMs consider it preferred, and two DMs consider it low preferred. Furthermore, the linguistic opinions of each expert can be transformed into TFNs according to Table 1, as shown in the following.

$$\begin{aligned}{E_{\text{5}}}&=\,{E_{\text{7}}}\,=\,{E_{{\text{1}}0}}={\text{ }}\left( {0.{\text{7}},{\text{ }}0.{\text{9}},{\text{ }}0.{\text{9}}} \right),{E_{\text{1}}}\,=\,{E_{\text{3}}}\,=\,{E_{\text{8}}}={E_{{\text{12}}}}=\left( {0.{\text{5}},{\text{ }}0.{\text{7}},{\text{ }}0.{\text{9}}} \right),\\ {E_{\text{2}}}&=\,{E_{\text{4}}}\,=\,{E_{{\text{11}}}}=\left( {0.{\text{3}},{\text{ }}0.{\text{5}},{\text{ }}0.{\text{7}}} \right),{E_{\text{6}}}\,=\,{E_{\text{9}}}=\left( {0.{\text{1}},{\text{ }}0.{\text{3}},{\text{ }}0.{\text{5}}} \right)\end{aligned}$$

Then, according to Eq. (4), we can obtain the aggregate fuzzy weight W₁=(α₁, β₁, γ₁) of criteria c₁, there exists.

$${\alpha _{\text{1}}}\,=\,{\text{min}}\left( {0.{\text{5}},0.{\text{3}},0.{\text{5}},0.{\text{3}},0.{\text{7}},0.{\text{1}},0.{\text{7}},0.{\text{5}},0.{\text{1}},0.{\text{7}},0.{\text{3}},0.{\text{5}}} \right)\,=\,0.{\text{1}}$$

$${\beta _{\text{1}}}=\left( {{\text{1}}/{\text{2}}} \right) \times \sum (0.{\text{7}}\,+\,0.{\text{5}}\,+\,0.{\text{7}}\,+\,0.{\text{5}}\,+\,0.{\text{9}}\,+\,0.{\text{3}}\,+\,0.{\text{9}}\,+\,0.{\text{7}}\,+\,0.{\text{3}}\,+\,0.{\text{9}}\,+\,0.{\text{5}}\,+\,0.{\text{7}})\,=\,0.{\text{633}}$$

$${\gamma _{\text{1}}}\,=\,{\text{max}}\left( {0.{\text{9}},{\text{ }}0.{\text{7}},{\text{ }}0.{\text{9}},{\text{ }}0.{\text{7}},{\text{ }}0.{\text{9}},{\text{ }}0.{\text{5}},{\text{ }}0.{\text{9}},{\text{ }}0.{\text{9}},{\text{ }}0.{\text{5}},{\text{ }}0.{\text{9}},{\text{ }}0.{\text{7}},{\text{ }}0.{\text{9}}} \right)\,=\,0.{\text{9}}$$

Thus, we obtain W₁=(0.1, 0.633, 0.9). The aggregated fuzzy weights of the other remaining criteria can be obtained by using the same way.

Next, in order to realize the aggregation of experts’ ratings over the three alternatives, we convert the linguistic ratings into TFNs according to Table 1 and Eq. (4). Subsequently, we use Eqs. (6) and (7) to normalize the aggregated fuzzy DM according to the positive or negative influencing factor. Furthermore, the obtained weighted normalized DM is determined using Eq. (8) based on the criteria weights and the normalized DM. Based on this, we can easily calculate FPIS and FNIS using Eqs. (9) and (10), respectively. Then, we use Eqs. (11) and (12) to calculate the desired distances $D_{i}^{+}$ and $D_{i}^{ - }$, as shown in Table 5. At the same time, the CC_i of each state in Table 5 is determined by using Eq. (13).

Table 5 Final ranking of the feasible States.

As shown in Table 5, the ranking results over all states in in resolving disputes in livelihood vulnerability assessment of flood by using F-TOPSIS is determined based on CC_i values, that is to say, s₁≻s₂≻s₃.

Triangular fuzzy stability definition in GMCR paradigm

Basic definitions of GMCR with TFNs

Within the GMCR with TFNs, compared to existing preference ranking approaches, F-TOPSIS method provides a new way to obtain DM’s real preference ranking in complex uncertain environments. Thus, the next work is to design stability definitions in GMCR with TFNs based on DM’s preference ranking, which can provide theoretical support for resolving conflicts.

To better character DM’s movement in complex uncertain environments, two novel concepts of reachable list based on TFNs are first given as follows.

Definition 10

Triangular fuzzy unilateral reachable list (TFURL): The reachable set in GMCR with TFNs from s for DM k is:

$$TF{R_k}(s)=\{ {s_1} \in S,(s,{s_1}) \in {A_k}\}$$

(14)

It is evident any movement of DM k with any step distance in TFR_k(s) is called triangular fuzzy unilateral movement.

Definition 11

Triangular fuzzy unilateral improvement list (TFUIL): The improved reachable set in GMCR with TFNs from s for DM k is:

$$TFR_{k}^{+}(s)=\{ {s_1} \in TF{R_k}(s),{s_1}{ \succ _k}s\}$$

(15)

It is evident any movement of DM k with any step distance in $TFR_{k}^{+}(s)$ is called triangular fuzzy unilateral improvement movement.

Note that the difference between Definitions 10 − 11 in GMCR with TFNs and Definitions 1 − 2 in traditional GMCR is that the preference degree over states in Definitions 10 − 11 is determined by using proposed F-TOPSIS method. Furthermore, conflict problems typically involve two or more DMs with different interest goals and wishes. Thus, the unilateral reachable set for a coalition H in GMCR with TFNs is rewritten as TFR_H(s)=$TF{R_{N - \{ k\} }}$(s).

Remark 2

It is obvious that TFR_H(s) in GMCR with TFNs is not involved improvement movement between states. Thus, TFR_H(s) is completely similar to R_H(s) in GMCR that described in Definition 5.

Definition 12

Let a coalition H⊆N and a state s∈S. Thus, $TFR_{H}^{+}(s)$ in GMCR with TFNs is shown as follows:

(1) If $k \in H$ and ${s_1} \in TFR_{k}^{+}(s)$, then ${s_1} \in TFR_{H}^{+}(s)$ and $k \in \Omega _{H}^{+}(s,{s_1})$.

(2) If $k \in H$, ${s_1} \in TFR_{H}^{+}(s)$, and ${s_2} \in TFR_{k}^{+}(s)$, then we obtain $\Omega _{H}^{+}(s,{s_1}) \ne \{ k\}$,${s_2} \in TFR_{H}^{+}(s)$ and $k \in \Omega _{H}^{+}(s,{s_2})$.

where $\Omega _{H}^{+}(s,{s_1})$ be the set of all last DMs in a legal sequence of triangular fuzzy unilateral improvement movement.

Triangular fuzzy stability definition of GMCR

Definition 13

Within GMCR with TFNs, a state s is TFNash for DM k iff $HMR_{k}^{+}(s)=\emptyset$, which represented as by using symbolic language s∈$S_{k}^{{TFNash}}$.

According to TFNash, there exit two different conflict situations, one is that for DM k, there are no other reachable states. Another is that although there are reachable states here, and the preference degree of all reachable states is not preferred than the initial state for DM k, which means that $TFR_{k}^{+}(s)=\emptyset$. Furthermore, when a focus state is considered to be TFNash stability for all DMs, the state is the global TFNash stable solution. As for considering the value of steps of DM’s movement between states, TFNash stability is typical one-step distance movement stability without taking account of opponents’ preferences.

Definition 14

Within GMCR with TFNs, a state s is TFGMR for DM k iff s₁∈$TFR_{k}^{+}(s)$, there exists another state s₂∈$TF{R_{N - \{ k\} }}({s_1})$ and possesses s≻_ks₂, which is represented by using symbolic language s∈$S_{k}^{{TFGMR}}$.

According to TFGMR, when DM k leaves from initial state to another state s₁, in the meanwhile, the opponent DMs $N - \{ k\}$ will move from state s₁ to s₂ to impose sanctions without considering personal interest gain or loss. Furthermore, DM k does not prefer state s₂ to initial state. Based on above analysis, s is TFGMR stability for DM k. Furthermore, when a focus state is considered to be TFGMR stability for all DMs, and the state is the global TFGMR stable solution. As for considering the value of steps of DM’s movement between states, TFGMR is two-step distance movement stability without taking account of opponents’ preferences.

Definition 15

Within GMCR with TFNs, a state s is TFSMR for DM k iff s₁∈$TFR_{k}^{+}(s)$, there exists another state s₂∈$TF{R_{N - \{ k\} }}({s_1})$ and possesses s≻_ks₂, and for any state of s₃∈TFR_k(s₂) and possesses s≻_ks₃, which represented as by using symbolic language s∈$S_{k}^{{TFSMR}}$.

TFSMR stability definitions are developed and expanded from the TFGMR stability definitions. Specifically, the first two steps in two definitions are perfectly same. The difference is that DM k in TFSMR will consider counterattack when considering opponent’s sanction. Based on the above analysis, the initial state is TFSMR stability for DM k. Furthermore, when a focus state is considered to be TFSMR stability for all DMs, and the state is the global TFSMR stable solution. As for considering the value of steps of DM’s movement between states, TFSMR is three-step distance movement stability without taking account of opponents’ preferences.

Definition 16

Within GMCR with TFNs, a state s is TFSEQ for DM k iff s₁∈$TFR_{k}^{+}(s)$, there exists another state s₂∈$TFR_{{N - \{ k\} }}^{+}({s_1})$ and possesses s≻_ks₂, which represented as by using symbolic language s∈$S_{k}^{{TFSEQ}}$.

TFSEQ stability is similar to the TFGMR stability. The difference between two new stability definitions is that in the second step in TFSEQ stability when DM k makes a movement between states s and s₁, the opponent DMs of focus DM k just consider the rational sanctions, represented as s₂∈$TFR_{{N - \{ k\} }}^{+}({s_1})$. The result is that state s₂ is not preferred to state initial state s for DM k. Furthermore, when a focus state is considered to be TFSEQ stability for all DMs, and the state is the global TFSEQ stable solution. As for considering the value of steps of DM’s movement between states, TFSEQ is two-step distance movement stability without taking account of opponents’ preferences.

Remark 3

If the conflicts just include two different DMs, a coalition $N - \{ k\}$ in Definitions 13 − 16 have been changed a single DM l. At this point, Definitions 13 − 16 are defined in the conflicts with two DMs.

Remark 4

A state is triangular fuzzy equilibrium (TFE) iff the state is triangular fuzzy stable for each DM based on one type of triangular fuzzy stability definitions.

Next, Theorem 1 shows the logical relationship among triangular fuzzy stability definitions.

Theorem 1

Within GMCR with TFNs, the relationships among new proposed triangular fuzzy stability definitions are:

$$S_{k}^{{TFNash}} \subseteq S_{k}^{{TFSMR}} \subseteq S_{k}^{{TFGMR}}$$

(16)

$$S_{k}^{{TFNash}} \subseteq S_{k}^{{TFSEQ}} \subseteq S_{k}^{{TFGMR}}$$

(17)

Proof

First, if a given state is TFNash stable, which means that $TFR_{k}^{+}(s)=\emptyset$. Second, for each state s₁∈$TFR_{k}^{+}(s)$, there exists s₂∈$TFR_{{N - \{ k\} }}^{+}({s_1})$ and possesses s≻_ks₂, and for s₃∈TFR_k(s₂) and possesses s≻_ks₃. Since $TFR_{{N - \{ k\} }}^{+}({s_1}) \subseteq TF{R_{N - \{ k\} }}({s_1})$, it follows that for state s₁∈$TFR_{k}^{+}(s)$, there exists s₂∈$TF{R_{N - \{ k\} }}({s_1})$ and possesses s≻_ks₂, and for any state of s₃∈TFR_k(s₂) and possesses s≻_ks₃, which represents that s is TFSMR stable. Thirdly, we suppose that a given state s is TFSMR stable, we obtain that for each state s₁∈$TFR_{k}^{+}(s)$, there exists s₂∈$TF{R_{N - \{ k\} }}({s_1})$ and possesses s≻_ks₂, which represents that a given state s is TFGMR stable. Thus, the interrelationships of (16) is proven. Finally, as for Eq. (17), the proof process is similar to interrelationships of (16), so it will not be repeated. □

Finally, Theorem 1 is described by using Venn diagram, as shown in Fig. 3.

Fig. 3

Interrelationships among the triangular fuzzy stability concepts in GMCR.

Conflict analysis of disputes in livelihood vulnerability assessment of flood in Yangtze river basin of China

Backgrounds

Since records began, the impact of natural disasters has significantly increased, causing serious economic losses and human injuries. Floods are one of the most common natural disasters, closely related to rivers and rainfall, and can be seen in almost any country on Earth. According to the latest statistics from the United Nations, approximately 150,000 people worldwide become homeless each year due to floods. This is especially true for China, where the Yangtze River basin brings abundant water resources but also causes serious flood disasters. This not only has a huge adverse impact on the normal production and life of the local area but also brings about the safety of people’s lives. The main reasons for the flooding of Yangtze River basin include natural and human factors. Among them, natural factors include the influence of southwest and southeast monsoons, mountainous and windward slopes at the junction of rivers, and low and flat terrain in the Yangtze River basin with poor drainage. Human factors include deforestation leading to severe soil and water flow, reclaiming land from lakes, and the Yangtze River basin is a region with concentrated agriculture, developed industry, and commercial trade in China. The above described has increased the vulnerability of the affected areas and also intensified the severity of the disaster. However, after the floods in the Yangtze River basin, the assessment methods established using precise digital data faced significant challenges in assessing the livelihood vulnerability of homeless persons. The severity of disasters, the lack of logistical support, and the damage to infrastructure all result in traditional survey methods being unable to directly obtain accurate numeric data from the affected people to carry out analysis work. Thus, the new proposed GMCR is used to resolve disputes in livelihood vulnerability assessment of flood in Yangtze River basin of China from qualitative analysis perspective. Finally, the sketch map of disputes in livelihood vulnerability assessment of flood in Yangtze River basin is shown in Fig. 4.

Fig. 4

The sketch map of disputes in livelihood vulnerability assessment of flood in Yangtze River basin.

Basic components of GMCR

(1) According to backgrounds of disputes in livelihood vulnerability assessment of flood in Yangtze River basin, DMs and their options are summarized as follows: Local government (LG): Officials are primarily responsible for supervising and managing local affairs, including infrastructure construction and maintenance, as well as policies and planning in flood management. They need to ensure that the community’s infrastructure can withstand the impact of floods, while also considering how to restore normal community operations after a flood event quickly. Water Resources Bureau (WRU): Management personnel are responsible for developing and implementing disaster management strategies. They need to ensure effective coordination of resources and protection of residents’ lives and property during flood events. Local residents (LR): During the flood period, the expectations of the people mainly focused on flood control and disaster relief and support and assistance from all sectors of society. Through the joint efforts and care of society, we can effectively respond to the challenges brought by flood disasters, help the affected people rebuild their homes, and restore production and life. Thus, the options of each DM are shown in Table 6.

Table 6 The options of each DM in disputes in livelihood vulnerability assessment of flood in Yangtze river basin.

(2) Feasible states. As discussed in backgrounds of disputes in livelihood vulnerability assessment of flood in Yangtze River basin, there exists three DMs and seven options, the options may be selected either “Y” or “N” for a DM. At the same time, “−” is used to represent the DM’s choice, which may be “Y” or “N”, as shown in Table 7. From the mathematical perspective, the conflict will form 2⁷=128 generated states. However, in practical problems, for example, DM 2 cannot select three options at the same time, and thereby form 9 feasible states.

Table 7 States disputes in livelihood vulnerability assessment of flood in Yangtze river basin.

(3) Directed graph. The integrated graph of disputes in livelihood vulnerability assessment of flood in Yangtze River basin is shown in Fig. 5.

Fig. 5

The directed graph of disputes in livelihood vulnerability assessment of flood in Yangtze River basin.

As shown in Fig. 5, there exists nine different vertexes, which are determined by the corresponding states. Three types arcs with different characteristic represent movment controlled by different DM. The directed graph of disputes in livelihood vulnerability assessment of flood in Yangtze River basin includes reversible movement and irreversible movement.

Conflict analysis result of the disputes in livelihood vulnerability assessment of flood in Yangtze river basin

Within the traditional GMCR, the existing preference ranking method cannot obtain preference ranking over all states in complex uncertain environments. F-TOSIS can be used to rank all states in complex environments based on uncertain preferences, which enriches the preference ranking methods in GMCR. In flood response, LG and WRU play a crucial role, and their responsibilities cover multiple aspects such as organizing, coordinating, supervising, and emergency response for flood prevention and control as well as reduce the livelihood damage of residents caused by floods. Thus, LG and WRU play a stakeholder and manager role, which means that LG and WRU have crisp preferences over all feasible states. The preference ranking for both DM LG and WRU can be determined by using direct method. Thus, the preferences for DM LG and WRU are s₇ ≻ s₃ ≻ s₄ ≻ s₈ ≻ s₅ ≻ s₁ ≻ s₂ ≻ s₆ ≻ s₉ and s₉ ≺ s₈ ≺ s₇≺ s₃≺ s₅≺ s₂ ≺ s₁ ≺ s₄≺ s₆, respectively. As a participant in the conflict, due to the lack of sufficient professional knowledge in flood control and reducing livelihood vulnerability, DM LR’s preferences over states are uncertain preference, and it is represented by using TFNs. Thus, F-TOPSIS is used to determine DM LR’s preference. Note that the calculation process is similar in analysis process in subsection 3.2. To save space of paper, the calculation is omitted. Thus, the preference ranking for DM LR is s₉ ≺ s₈≺ s₇≺ s₃≺ s₅≺ s₂≺ s₁≺ s₄≺ s₆. Obviously, if we wish to obtain triangular fuzzy stability analysis result in conflict, the preference ranking for each DM is an important basic condition. Thus, based on Definitions 13 − 16, the triangular fuzzy stability analysis results is shown in Table 8.

Table 8 The triangular fuzzy stability analysis result in disputes in livelihood vulnerability assessment of flood in Yangtze river basin.

As shown in Table 8, TFE represents the state is stable for all DMs in disputes in livelihood vulnerability assessment of flood in Yangtze River basin based on a certain triangular fuzzy stability definitions. Obviously, two states s₄ and s₉ are considered as potential TFE equilibrium based on four triangular fuzzy stability definitions. However, the movement between other states and state s₉ is just influenced by DM WRU’s preference relations. At the same time, state s₉ is composed by “− − − − Y − −”, which means that the state s₉ does not have practical guidance significance. Based on the above analysis, state s₄ is only triangular fuzzy stability stable for resolving disputes in livelihood vulnerability assessment of flood in Yangtze River basin. To be specific, flood prevention and disaster relief is one of the basic responsibilities of the LG. LG needs to take a series of measures, including planning and design, construction of flood control facilities, and improvement of flood control laws and regulations, to ensure that the region can withstand the impact of floods and provide timely rescue. WRU is subject to the management of the LG and aims to improve the livelihood level of local residents through strengthening early warning and response mechanisms, on-site guidance and emergency rescue, as well as post disaster reconstruction. When facing flood disasters, LR can provide strong support to the LG and WRU through timely reporting of the disaster situation, participating in post disaster reconstruction, and participating in flood prevention propaganda. It can protect their own and others’ safety during disasters and jointly improve their livelihood level.

Comparative analysis among three different types of GMCR

The novel GMCR proposed in this paper allows DMs use TFNs to deal with the uncertain preferences of experts in conflict analysis process. Thus, the comparison among GMCR with TFNs, traditional GMCR²³, and fuzzy GMCR⁴³ have been carried out from there aspects, as shown in Table 9.

Table 9 Comparative analysis three different types of GMCR.

As shown in Table 9, GMCR with TFNs, traditional GMCR, and fuzzy GMCR have proposed a set of stability definition to ensure the completeness of stability analysis results. Compared to traditional GMCR, GMCR with TFNs and fuzzy GMCR can be used to obtain optimal conflict resolution in complex uncertain environment. Compared with traditional GMCR and fuzzy GMCR, F-TOSIS is proposed to obtain preference ranking over all states for each DM within GMCR with TFNs.

Next, to compare three different types of GMCR from a numerical perspective, we apply them to the livelihood vulnerability assessment of flood in the Yangtze River basin to obtain equilibrium. More importantly, the results of stability analysis can clearly and intuitively show the difference between different types of conflict analysis graph models in solving practical problems, as shown in Table 10.

Table 10 Comparison of the stability results among three different GMCR in the disputes in livelihood vulnerability assessment of flood in Yangtze river basin.

As shown in Table 10, from a numerical analysis perspective, the stability results of the disputes in livelihood vulnerability assessment of flood in Yangtze River basin by using three different types of GMCR are clearly difference from each other. First, traditional GMCR is difficult to apply to conflicts because DM’s preferences are uncertain preferences. Thus, the stability analysis result obtained by crisp graph model is none. In other words, the new proposed method can be used to resolve conflicts in uncertain environments. Second, the stability analysis results of the disputes are considered to be same by using fuzzy graph model and GMCR with TFNs at the same time. However, compared to fuzzy graph model, GMCR with TFNs provides a new preference elicitation method, which provides new insights and indicate greater practicality. Third, compared to other two different GMCR, only the GMCR with TFNs can analyze the conflicts that occur when DMs show triangular fuzzy preferences to resolve conflicts, showing its excellent applicability.

Implications of the research

As one of the most powerful and strategically supportive regions in China, the Yangtze River basin is facing an increasingly severe crisis of floods and waterlogging due to human factors and special natural conditions. The conflict analysis of disputes in livelihood vulnerability assessment of flood in the region will help to understand the differences in social vulnerability policies of various DMs, as well as the impact of factors such as population, economy, infrastructure, and agricultural development on the social vulnerability of flood disasters in the region. By making corresponding modifications for DMs with high exposure and sensitivity, while improving the adaptability of flood prevention, we can improve the ecological environment conditions and reduce the economic losses caused by flood disasters.

LG can improve engineering measures such as improving drainage systems, increasing flood discharge channels, improving infrastructure construction, as well as non-engineering measures such as drawing flood risk maps, defining risk areas, strengthening flood forecasting, dispatching, warning and other flood control command system construction by improving emergency plans for defense against exceeding standards, in order to better guarantee residents’ livelihoods in all aspects. WRU greatly cooperates with the government’s emergency activities through material and manpower support to improve the emergency response to floods. In addition, WRU dispatches professionals to ensure the health and hygiene of residents in disaster stricken areas. LR need to have a deep understanding and learn about disaster prevention and mitigation knowledge related to floods, enhance emergency awareness, and strengthen their awareness of flood prevention, disaster reduction, and risk avoidance. Moreover, LR must have an understanding of the possibility of disasters occurring in their area and make corresponding emergency preparations, such as preparing emergency drugs and food. By making sufficient preparations before a disaster, one can remain calm and composed in the face of danger, promote the development of things in a positive direction, and ensure the smooth progress of disaster relief work.

Conclusions and future works

In recent years, the frequency and intensity of extreme weather such as rainstorm have significantly increased, and the flood problem has become increasingly serious, which has become an important factor affecting the livelihood of residents. Particularly, The Yangtze River basin is located in central China, with a vast hinterland, fertile land, abundant resources, and developed agriculture. It is the region with the best natural geographical foundation in China. Therefore, due to its unique natural geographical environment, the Yangtze River basin is facing a very severe crisis of flood disasters. Flood disasters have become one of the important factors affecting residents’ livelihood security. This article has important theoretical and practical significance in exploring the impact of flood disasters on residents’ livelihoods from the perspective of vulnerability by using GMCR with TFNs. The contributions of this paper are summarized as follows:

(1) GMCR with TFNs is proposed to resolve disputes in livelihood vulnerability assessment of flood, which can analyze the impact of floods in the Yangtze River basin on the livelihoods of local residents, explore the livelihood changes, vulnerability levels, and regulation strategies of different stakeholders, and provide scientific references for reducing the livelihood damage of residents caused by floods and improving their ability to cope and adapt to disasters.

(2) To overcome the limitations of preference ranking methods in traditional GMCR that are difficult to apply in uncertain environments and to adapt to the characteristics of disputes in livelihood vulnerability assessment of flood, F-TOPSIS is designed into GMCR to obtain DM’s preference relations.

(3) A set of stability definitions with TFNs is first proposed to GMCR, which has driven the development of GMCR in uncertain environments. Compared to basic stability definitions in traditional GMCR, the proposed stability definitions have broader applicability and can solve conflicts in multiple fields.

(4) Conflict analysis result of the disputes in livelihood vulnerability assessment of flood in Yangtze River basin attempt to analyze the impact of flood disasters on the livelihood changes and vulnerability of various stakeholders from a quantitative analysis perspective, which can provide support for the early development of targeted flood prevention plans and reduce the impact of flood disasters on residents’ livelihoods.

(5) This article not only proposes regulatory measures to enhance the livelihood security of urban residents and reduce their livelihood vulnerability, but also provides certain supplements and references for flood disaster management in the Yangtze River basin.

In the future works, we propose some interesting and important research avenues based on the research conclusion in this paper.

(1) We will integrate consensus theory into the GMCR to character the real-life DM’s preferences influenced by disputes in livelihood vulnerability assessment of flood, which can ensure that the conflict analysis results are more authentic and reliable.

(2) By conducting in-depth analysis of DMs’ preferences and conflict backgrounds, it is possible to enhance the validity of weight values in GMCR by using F-TOPSIS.

(3) To overcome the different preference expression type of DMs and complex conflict backgrounds of disputes in livelihood vulnerability assessment of flood, using different types of preferences regarding the DM’s evaluation habits should be taken into account in GMCR.

(4) When conflict occurs in different groups, individuals within the organization may have different preferences towards the feasible states due to differences in knowledge, experience, and other factors, which make it difficult to obtain clear preference information of organizations. Thus, the composite DMs that represent a set of stakeholders or organizations with common interest goals are proposed in GMCR. Furthermore, we will apply the consensus model to the achievement of individual unity of opinion within the CDMs in order to facilitate the subsequent conflict analysis process by using grey relational analysis, social trust network, DeGroot opinion evolution, and k-means clustering method.