Shapley Index Research Articles

An improved case-based reasoning approach for sustainable rural development applied to strategic responses

Sustainable development has become a worldwide consensus in recent years. It has become increasingly important to study how to make effective management decisions in rural areas to achieve sustainable development. In this paper, we develop a new approach to strategic responses for sustainable rural development by using case-based reasoning (CBR), the bi-direction Choquet integral (BDCI), and the Shapley index (SI). First, to deal with the complexity of sustainable rural development, we develop an ontology-based sustainable rural development domain model. Second, we propose a case retrieval algorithm based on BDCI to improve the case retrieval method used in CBR to solve the problem of the relationship between the features. Third, we propose a case reuse method that is inspired by SI, in which each solution is evaluated relative to its value, and the implementation step for the response strategy is determined based on its priority. The proposed strategy response method for sustainable rural development is demonstrated through a case study and its effectiveness is demonstrated empirically.

Determinants of health poverty vulnerability in rural areas of Western China in the post-poverty relief era: an analysis based on the Anderson behavioral model

BackgroundAlthough China has eliminated absolute poverty, the effects of sickness still pose a threat to the prospect of returning to poverty in western rural areas. However, poverty governance extends beyond solving absolute poverty, and should enhance the family’s ability to resist risks, proactively identify the existence of risks, and facilitate preventive measures to reduce the probability of falling into poverty again. This study aimed to assess the health poverty vulnerability of rural households in western China and decompose its determinants.MethodsBased on survey data from 2022, the three-stage feasible generalized least squares method was used to calculate the health poverty vulnerability index. Then, Anderson’s health behavior theory model was extended to analyse various influencing factors using binary logistic regression, and the contribution of each influencing factor was decomposed using the Shapley index. Finally, Tobit regression and the censored least absolute deviations estimation (clad) method were used to test the model’s robustness.ResultsA total of 5455 families in the rural Ningxia region of western China were included in the study. The health poverty vulnerability index of the sample population in 2022 was 0.3000 ± 0.2223, and families with vulnerability ≥0.5 accounted for 16.9% of the sample population. From the Anderson behavioral model, the three models including propensity, enabling, and demand factors had the best fit, and the AIC and BIC values were the smallest. The Shapley decomposition showed that the dimensions of the propensity factor, number of residents, age and educational level of the household head, and dependency ratio were the most important factors influencing vulnerability to health poverty. Tobit regression and the clad method proved the reliability of the constructed model through a robustness test.ConclusionRural areas still face the risk of becoming poor or falling into poverty owing to residents’ health problems. Health poverty alleviation should gradually change from a focus on treatment to prevention, and formulate a set of accurate and efficient intervention policies from a forward-looking perspective to consolidate the results of health poverty alleviation and prevent widescale poverty return.

Isolation Forest With Exclusion of Attributes Based on Shapley Index

Isolation Forest With Exclusion of Attributes Based on Shapley Index

The cross-efficiency evaluation model with judgmental information: Cooperative game and optimal coordination perspective

In the actual efficiency evaluation process, it frequently occurs that the quantitative input and output data of each decision-making unit is difficult to obtain due to the limitation in data measurement and the absence of prior information. Instead, the form of judgment information can be selected to express the input–output state of the evaluated decision-making units compared with the reference level. Traditional cross-efficiency evaluation methods do not involve this case and cannot be well used in this situation. Based on the above problems, a multiplicative cross-efficiency evaluation model with judgmental information is constructed, and the optimal adjustment model is provided by introducing deviation variables to deal with the inconsistent issue of data features. In addition, for the case of multiple optimal solutions, the maximizing and minimizing secondary objective models of the multiplicative cross-efficiency are established respectively to reasonably identify the corresponding optimal weights. Simultaneously, in order to effectively depict the interaction between decision-making units, the characteristic functions of JM-cross efficiency cooperative game and the multiplicative Shapley value are established based on benevolent and aggressive cross-efficiency matrices. Moreover, in order to avoid the extreme ranking results that are oriented by high impact decision-making units, a series of optimization coordination models are developed, and an optimal efficiency ranking vector generation method with strong applicability and decision flexibility is proposed. The optimal efficiency ranking vector not only creates a precise match with the given Shapley index of the multiplicative cross-efficiency game, but also bounds between the minimum and the maximum average efficiency. Finally, the validity and rationality of the discussed models are illustrated by a concrete example.

Evaluation of China’s Targeted Poverty Alleviation Policies: A Decomposition Analysis Based on the Poverty Reduction Effects

To achieve comprehensive poverty alleviation and the establishment of a “moderately prosperous society” in China, it is crucial to evaluate the targeted poverty alleviation (TPA) policies. In this study, China’s poverty alleviation statistics and the Foster-Greene-Thorbecke (FGT) indices are used to measure the poverty reduction effects of the TPA policies. A panel regression model is applied to analyze the poverty reduction mechanism while the Shapley index decomposition method is used to analyze poverty reduction effects in terms of income growth and the income gap adjustment. The paper concludes that the poverty breadth index (H index), poverty depth index (PG index), and poverty intensity index (SPG index) from 2013 to 2019 show a significant decline overall. This indicates that the poverty reduction effect of the TPA policies is significant. In addition, the regression analysis shows that the implementation of TPA policies can significantly increase the income level of residents and narrow the income gap among residents in rural areas. Results of the Shapley index decomposition analysis revealed that the income growth effect and income gap adjustment effect accounted for 92.78% and 7.22% of the poverty reduction effects, respectively. So the focus of future poverty alleviation work is to combine the rural revitalization strategy and to continue increasing the income level and the income growth rate of poor groups, which will enhance the ability of impoverished residents to increase their income, further contributing to the alleviation of poverty.

Application of Choquet Integral-Fuzzy Measures for Aggregating Customers’ Satisfaction

Choquet integral is a type of aggregation operator that is commonly used to aggregate the interrelated information. Nowadays, this operator has been successfully embedded with fuzzy measures in solving various evaluation problems. Inspired from this new development, this paper aims to introduce a combined Choquet integral-fuzzy measures (CI-FM) operator that uses the Shapley value standard and interaction index to deal with the interactions between elements of information. The proposed operator takes into account not only the importance of elements or their ordered positions but also the interaction among criteria during the evaluation process. A case of customers’ satisfaction over two fast restaurants in Malaysia is presented to illustrate the application of the proposed aggregation operator. Based on three customers’ satisfaction criteria, restaurant 1 and restaurant 2 received CI-FM scores of 0.711011 and 0.704945, respectively. Interestingly, the criterion “services” constantly received the highest rating across both restaurants. In addition, the proposed aggregation operator successfully identified which restaurant is superior in the eyes of customers. Finally, this study offers some research ideas for the future.

Classification and influencing factors of rural elderly's vulnerability to health-related poverty in central and western regions of China

ObjectiveTo calculate the health poverty vulnerability index of elderly households in rural areas of central and western China, and then to classify these samples, lastly to decompose their influencing factors. MethodsFirst, based on survey data in 2018, the three-stage feasible generalized least squares was used to calculate the health poverty vulnerability index of elderly households, and then combined with whether the household income was below the poverty line and whether the family was healthy poverty vulnerability, the sample households were divided into four categories, and then used multiple unordered logistic regression to analyze various types of influencing factors, and finally used the Shapley index to decompose the contribution of each influencing factor. ResultsThe average vulnerability of health poverty was 0.5979 ± 0.25199, with 1169 households greater than or equal to 0.5, accounting for 63.26%; the number of households stuck in poverty, temporary poverty, potential poverty, and escaped from poverty were 489, 300, 680, and 379 households, accounting for 26.46%, 16.23%, 36.80%, and 20.51% of the total sample; compared with escaped from poverty families, the three variables of marital status, the number of chronically ill patients, and the number of annual hospitalizations were the common influencing factors of other three types families; The Shapley decomposition showed that the interviewees’ education level and family members engaged in non-agricultural work have contributed significantly to the three types, however two indicators: time required to visit a medical institution and self-assessment of health status of the main interviewees showed great differences in different types of families. ConclusionRural elderly households have a high level of vulnerability to health poverty; potential poverty households and persistent poverty households account for a large proportion, and continuous intervention should be carried out; it is necessary to unify the implementation of basic poverty alleviation work, but also to enhance refined management capabilities and adopt differentiated intervention measures.

Shapley Distance and Shapley Index for Some Special Graphs

The Shapley distance in a graph is defined based on Shapley value in cooperative game theory. It is used to measure the cost for a vertex in a graph to access another vertex. In this paper, we establish the Shapley distance between two arbitrary vertices for some special graphs, i.e., path, tree, cycle, complete graph, complete bipartite, and complete multipartite graph. Moreover, based on the Shapley distance, we propose a new index, namely Shapley index, and then compare Shapley index with Wiener index and Kirchhoff index for these special graphs. We also characterize the extremal graphs in which these three indices are equal.

Granulized Z-VIKOR Model for Failure Mode and Effect Analysis

In this article, we have developed an improved failure mode and effect analysis (FMEA) model by leveraging the concepts of Z-number, rough number (RN), and probabilistic distance measure. Two new concepts, namely, double upper approximated rough number (DUARN) and granulized Z-number (gZN), a new scheme for measuring distance between two gZNs using weighted similarity and average linkage method, a new risk prioritization model, named, granulized Z-VIKOR, and a scheme for uncertainty assessment using box-plot are proposed. DUARN embodies the notion of double sided upper approximation of an ordinal decision class. gZN is developed using Z-number and DUARN. The distance between two gZNs is computed using the maximum entropy principle that captures the relationship between the A (opinion) and B (reliability of A) parts of gZN. The granulized Z-VIKOR involves synergistic integration of gZN and VIKOR. Both objective and subjective risk measures are computed and a combined risk measure is defined, which considers the interactions among the risk criteria using λ-Shapley index. Two case studies are conducted. Sensitivity and comparative analysis is carried out to demonstrate the applicability, effectiveness, and robustness of the proposed model, as well as its superiority to existing models.

Selection of product recycling channels based on extended TODIM method

The evaluation and selection of product recovery channels is a key decision issue in closed-loop supply chain management. Although existing research does not take into account the presence of inaccurate information and the influence of the decision maker’s psychological state during the selection of recycling channels, we study the issue of recycling channel selection under the consideration of inaccurate information and the influence of the decision maker’s psychological state. (1) First, we establish a new evaluation index system for recycling channels for policy makers. (2) We propose a TODIM (an acronym in Portuguese of Interactive and Multi-criteria Decision Making) method based on fuzzy evaluation and Shapley index for decision makers to imitate the psychological behavior characteristics of FMEA (Failure mode and effect analysis) team members. It can also be used to estimate the risk priority of several failure modes. The interval type 2 fuzzy is used to describe the uncertainty in the risk assessment process. (3) Finally, we obtain the practical significance of the method proposed in this paper through case studies, sensitivity analysis and comparative analysis of traditional methods. We also find the key impact factors in the choice of recycling channels.

Shapley and superShapley aggregation emerging from consensus dynamics in the multicriteria Choquet framework

We investigate a negotiation model for the progressive aggregation of interacting multicriteria evaluations. The model is based on a network of interacting criteria and combines the Choquet aggregation framework with the classical DeGroot’s model of consensus linear dynamics. We consider a set $$N = \{ 1,\ldots ,n \}$$ of interacting criteria whose single evaluations are expressed in some domain $${\mathbb {D}}\subseteq {\mathbb {R}}$$ . The pairwise interaction among the criteria is described by a complete graph with edge values in the open unit interval. In the Choquet framework, the interacting network structure is the basis for the construction of a consensus capacity $$\mu $$ , whose Shapley indices are proportional to the average degree of interaction between criterion $$i \in N $$ and the remaining criteria $$j \ne i \in N $$ . We discuss three types of linear consensus dynamics, each of which represents a progressive aggregation process towards a consensual multicriteria evaluation corresponding to some form of mean of the original multicriteria evaluations. All three models refer significantly to the notion of multicriteria context evaluation. In one model, the progressive aggregation converges simply to the plain mean of the original multicriteria evaluations, while another model converges to the Shapley mean of those original multicriteria evaluations. The third model, instead, converges to an emphasized form of Shapley mean, which we call superShapley mean. The interesting relation between Shapley and superShapley aggregation is investigated.

Strategic sustainable development selection using a new multi-criteria approach 'TOPSISIC': real case study

Ecology has recently been grasping for relevance in Tunisia. Additionally, the agglomerations are reflecting on the importance of sustainable development. This study carries out a quantitative study in the 'MONOPRIX' food store in Sfax. The data were collected using sustainable development strategies, which are the alternatives, as well as on the identification of the criteria. Subsequently, the classification and the choice of the most suited preferences to the decision-maker have been developed with the TOPSIS method. The results were not satisfied with the DM preferences. To satisfy this need, we have proposed to integrate the Choquet integral who considered the importation of the dependence relations and coalitions between the criteria by using the Shapley index for the reformulation of the mathematical formula in the ponderation step of the classic TOPSIS. This combination has resulted in a new TOPSISIC approach to outline a better strategy for the MONOPRIX food retailers.

Strategic sustainable development selection using a new multi-criteria approach 'TOPSISIC': real case study

Ecology has recently been grasping for relevance in Tunisia. Additionally, the agglomerations are reflecting on the importance of sustainable development. This study carries out a quantitative study in the 'MONOPRIX' food store in Sfax. The data were collected using sustainable development strategies, which are the alternatives, as well as on the identification of the criteria. Subsequently, the classification and the choice of the most suited preferences to the decision-maker have been developed with the TOPSIS method. The results were not satisfied with the DM preferences. To satisfy this need, we have proposed to integrate the Choquet integral who considered the importation of the dependence relations and coalitions between the criteria by using the Shapley index for the reformulation of the mathematical formula in the ponderation step of the classic TOPSIS. This combination has resulted in a new TOPSISIC approach to outline a better strategy for the MONOPRIX food retailers.

Consensus dynamics, network interaction, and Shapley indices in the Choquet framework

We consider a set $$N = \{ 1,\ldots ,n \}$$ of interacting agents whose individual opinions are denoted by $$ x_{i}$$ , $$i \in N $$ in some domain $${\mathbb {D}}\subseteq {\mathbb {R}}$$ . The interaction among the agents is expressed by a symmetric interaction matrix with null diagonal and off-diagonal coefficients in the open unit interval. The interacting network structure is thus that of a complete graph with edge values in (0, 1). In the Choquet framework, the interacting network structure is the basis for the construction of a consensus capacity $$\mu $$ , where the capacity value $$\mu (S)$$ of a coalition of agents $$S \subseteq N$$ is defined to be proportional to the sum of the edge interaction values contained in the subgraph associated with S. The capacity $$\mu $$ is obtained in terms of its 2-additive Mobius transform $$m_{\mu }$$ , and the corresponding Shapley power and interaction indices are identified. We then discuss two types of consensus dynamics, both of which refer significantly to the notion of context opinion. The second type converges simply the plain mean, whereas the first type produces the Shapley mean as the asymptotic consensual opinion. In this way, it provides a dynamical realization of Shapley aggregation.

An extended generalized TODIM for risk evaluation and prioritization of failure modes considering risk indicators interaction

Failure Mode and Effect Analysis (FMEA) is considered as a proactive risk prevention and control technique that has been widely applied to identify, assess and eliminate the risk of failure modes in various fields. Nevertheless, the interactions between risk indicators and decision maker’s psychological behavior characteristics are seldom considered simultaneously in the current FMEA method. In this article, we develop a hybrid FMEA framework integrating generalized TODIM (an acronym in Portuguese of Interactive and Multi-criteria Decision Making) method, Choquet integral and Shapley index to remedy this gap. In the proposed FMEA framework, fuzzy measures and Shapley index are used to model the interaction relationships among risk indicators and to determine the weights of these indicators. The extended generalized TODIM method with fuzzy measure and Shapley index is presented to simulate the psychological behavior characteristics of FMEA team members. It is also applied to calculate the risk priority of each failure mode. Trapezoidal Fuzzy Numbers (TrFNs) are adopted to depict the uncertainty in the risk evaluation process. Furthermore, a new risk evaluation information fusion with TrFNs-WAIA (weighted arithmetic interaction averaging operator of the trapezoidal fuzzy numbers) operator based on Shapley Choquet is developed to aggregate individual risk evaluation information of FMEA team member into a group risk evaluation matrix, which considers the potential correlations among these members. Finally, a practical example of FMEA problem is presented to demonstrate the application and feasibility of the proposed hybrid FMEA framework, and comparison and sensitivity studies are also conducted to validate the effectiveness of the improved FMEA approach.

The Coleman–Shapley index: being decisive within the coalition of the interested

The Coleman power of a collectivity to act (CPCA) is a popular statistic that reflects the ability of a committee to pass a proposal. Applying the Shapley value to that measure, we derive a new power index—the Coleman–Shapley index (CSI)—indicating each voter’s contribution to the CPCA. The CSI is characterized by four axioms: anonymity, the null voter property, the transfer property, and a property stipulating that the sum of the voters’ power equals the CPCA. Similar to the Shapley–Shubik index (SSI) and the Penrose–Banzhaf index (PBI), our new index reflects the expectation of being a pivotal voter. Here, the coalitional formation model underlying the CPCA and the PBI is combined with the ordering approach underlying the SSI. In contrast to the SSI, voters are ordered not according to their agreement with a potential bill, but according to their vested interest in it. Among the most interested voters, power is then measured in a way similar to the PBI. Although we advocate the CSI over the PBI so as to capture a voter’s influence on whether a proposal passes, our index gives new meaning to the PBI. The CSI is a decomposer of the PBI, splitting the PBI into a voter’s power as such and the voter’s impact on the power of the other voters by threatening to block any proposal. We apply our index to the EU Council and the UN Security Council.

A extended intuitionistic fuzzy Choquet integral correlation coefficient based on Shapley index in multi-criteria decision making

A extended intuitionistic fuzzy Choquet integral correlation coefficient based on Shapley index in multi-criteria decision making

On Using the Shapley Value to Approximate the Choquet Integral in Cases of Uncertain Arguments

In multicriteria decision making, the Choquet integral provides a commonly used method for the aggregation of the individual criteria satisfaction. Information about the criteria importance is expressed via a fuzzy measure on the set of criteria. One requirement in using the Choquet integral is to provide an ordering over the individual criteria satisfactions. When there exists some uncertainty regarding the criteria satisfactions, there typically exists some difficulty in providing this ordering. In an attempt to circumvent this difficulty, we suggest an approximation to the Choquet integral criteria aggregation that does not require an ordering. In this approximation, the criteria satisfactions are aggregated using as weights the Shapley index of the criteria. We provide an assessment of the merit of the approximation using the cardinality indices of the importance measure.

Energy-Related CO2 Emission in China’s Provincial Thermal Electricity Generation: Driving Factors and Possibilities for Abatement

China’s electricity sector mainly relies on coal-fired thermal generation, thus resulting that nearly 50% of China’s total CO2 emissions coming from the electricity sector. This study focuses on the provincial CO2 emissions from China’s thermal electricity generation. Methodologically, Index Decomposition Analysis (IDA), facilitated by the Shapley Index, is applied to discover the driving factors behind CO2 emission changes at the provincial level. In addition, the Slack-based Model (SBM) is used to identify which provincial power grids should be allocated with a higher (lower) CO2 reduction burden. The IDA results indicate that economic activity pushed the CO2 emissions up in all 30 provincial power grids, excluding Beijing and Shanghai; the carbon factor contributed to a decrease in the CO2 emissions in all 30 provincial power grids, with the exception of Jilin, Guangdong, and Ningxia; though the effect of energy intensity varied across the 30 provinces, it played a significant role in the mitigation of CO2 emissions in Beijing, Heilongjiang, Liaoning, Jilin, Shanghai, Anhui, and Sichuan. According to the SBM results, the lowest carbon shadow prices are observed in Yunnan, Shanghai, Inner Mongolia, Jilin, Qinghai, Guizhou, Anhui, and Ningxia. These provincial power grids, thus, should face higher mitigation targets for CO2 emissions from thermal electricity generation.

Measures of the Shapley index for learning lower complexity fuzzy integrals

The fuzzy integral (FI) is used frequently as a parametric nonlinear aggregation operator for data or information fusion. To date, numerous data-driven algorithms have been put forth to learn the FI for tasks like signal and image processing, multi-criteria decision making, logistic regression and minimization of the sum of squared error (SEE) criteria in decision-level fusion. However, existing work has focused on learning the densities (worth of just the individual inputs in the underlying fuzzy measure (FM)) relative to an imputation method (algorithm that assigns values to the remainder of the FM) or the full FM is learned relative to a single criteria (e.g., SSE). Only a handful of approaches have investigated how to learn the FI relative to some minimization criteria (logistic regression or SSE) in conjunction with a second criteria, namely model complexity. Including model complexity is important because it allows us to learn solutions that are less prone to overfitting and we can lower a solution’s cost (financial, computational, etc.). Herein, we review and compare different indices of model (capacity) complexity. We show that there is no global best. Instead, applications and goals (contexts) are what drives which index is appropriate. In addition, we put forth new indices based on functions of the Shapley index. Synthetic and real-world experiments demonstrate the range and behavior of these different indices for decision-level fusion.