Α-cut Levels Research Articles

Evaluating the Efficiencies of Logistics Centers with Fuzzy Logic: The Case of Turkey

The primary actor in today’s economic life, forming the backbone of the production-consumption cycle, is the distribution activities. Logistics centers (LCs) are organized areas where these activities are carried out together. Therefore, the efficiency and effectiveness of distribution activities are crucial for sustainability. This study incorporates fuzzy logic theory into the framework of data envelopment analysis (DEA) to measure the efficiency of LCs. Classical DEA assumes input and output data are precisely measured, making the efficiency scores unreliable and inconsistent when data precision is not always possible. The adoption of fuzzy logic is primarily to overcome possible uncertainties, errors, and ambiguities in data acquisition, preventing incorrect results. Hence, an approach assumes the data lie within specific intervals, was adopted to calculate the efficiencies of LCs based on α-cut levels. Officially obtained data on nine input and one output variable from twelve LCs operating in Turkey were used to calculate efficiency scores. As a result of the study, Köseköy/Izmit, Halkali/Istanbul, and Yenice/Mersin LCs were found to be fully efficient considering both lower and upper bound efficiencies. Moreover, the efficiency calculations using Fuzzy-DEA allowed for a more precise evaluation of LCs with high data sensitivity.

Uncertainty analysis of water quality in water distribution system

Abstract Water quality simulation is affected by uncertain parameters such as pipe roughness coefficients, chlorine bulk decay coefficients, and chlorine wall decay coefficients, which are usually considered to be fuzzy variables. The minimum and maximum nodal chlorine concentrations and water ages at each α-cut level were obtained by the genetic algorithm (GA) based on EPANET hydraulic and water quality simulation toolkit. The fuzziness of nodal chlorine concentrations and water ages were measured using the fuzziness measure (FM) proposed in this paper. The method was applied to four networks to analyze the fuzziness of nodal chlorine concentrations and water ages. The results indicated that the distribution of nodal chlorine concentrations does not follow typical trapezoid distribution, while the distribution of nodal water ages follows typical trapezoid distribution. In addition, the chlorine concentration and water ages of nodes farther from the source are affected by uncertain parameters to a greater extent. The greater demands of nodes lead to less effects of uncertain parameters on chlorine concentration, and greater effects of uncertain parameters on water ages. This study would help in analyzing the fuzziness of hydraulic and water quality simulation results in WDS under uncertain conditions.

An interval-based multi-objective robust design optimization for vehicle dynamics

This study presents an Interval-based Multi-objective Robust Design Optimization (IB-MORDO) algorithm applied to a vehicle dynamic problem. The proposed algorithm optimizes a full 15 degrees-of-freedom (15-DOF) vehicle model, subjected to a double-lane change (DLC) maneuver under random road profiles, to attain driver comfort and safety. This study does not make assumptions about uncertain parameter statistics; instead, the uncertainties are quantified using a non-probabilistic α-cut level interval analysis. These uncertainties are applied to the system parameters and design variables to ensure robust results. After the optimization process, the root mean square (RMS) vertical acceleration at the driver’s seat resulted in a robust solution of 1.041 m/s2 and a parameter interval radius (IR) equals to 0.631 m/s2, whereas the RMS lateral acceleration at the driver’s seat resulted in a solution of 1.908 m/s2 with an interval radius of 0.168 m/s2. Unlike the Robust Optimization, the algorithm proposed herein considers uncertainties at system parameters and design variables without assuming any statistical data. HIGHLIGHTS An Interval-based Robust Multi-objective Optimization procedure is proposed and tested on a 15-DOF vehicle model. Αn α-cut level methodology is used to deal with the uncertainty propagation. Resulted optimal suspension parameters minimize center and interval radius of driver’s vertical and lateral accelerations.

The attitude of MCDM approaches versus the optimization model in finding the safest shortest path on a fuzzy network

This paper examines the performances of the multi-criteria decision-making (MCDM) methods and optimization model in solving multi-attribute shortest path problems such as the safest shortest path under a fuzzy environment. To the best of the knowledge of the authors, this is the first study performing comparative analysis on finding the multi-attribute shortest path by employing well-known techniques in terms of computational effort and results in a fuzzy environment. To this end, the safest shortest path problem, where the risk and distance values concerning arcs on a directed network are defined as triangular fuzzy numbers, is handled. The solution process is carried out under two main headings: (i) To start the solution with MCDM methods, an auxiliary algorithm that constructs a fuzzy decision matrix is proposed. Then, Fuzzy-Technique for Order Preference by Similarity to Ideal Solution (F-TOPSIS), Fuzzy Simple Additive Weighting (F-SAW), and Fuzzy Evaluation Based on Distance from Average Solution (F-EDAS), that are fuzzy-based MCDM methods, are employed to rank the alternative paths. (ii) A multi-objective fuzzy optimization model is formulated, and the most reasonable paths are obtained considering different α-cut levels. Following that, comparative analysis is performed through a set of scenarios considering the different weights of the criteria to see the variability in the rankings. Besides, the addressed fuzzy-based MCDM methods are compared in terms of computational complexity. Overall, the main findings and managerial insights regarding the effectiveness and performance of the methods discussed in the solution process are provided.

Prediction and optimization of regional land-use patterns considering nonpoint-source pollution control under conditions of uncertainty

Socioeconomic development, leading to significant changes in land-use patterns, has further influenced the output of regional nonpoint-source (NPS) pollution. Multiple uncertainties exist in the processes of land-use changes and NPS pollution export. These uncertainties can deeply affect the management of regional land-use patterns and control of NPS pollution. In this research, an integrated land-use prediction and optimization (ILUPO) model based on system dynamics, export coefficient, interval linear programming, and fuzzy parameter programming models was proposed. The ILUPO model can provide future land-use patterns and NPS pollution loads, and also help optimize the patterns under multiple pollution reduction scenarios. Interval and fuzzy uncertainties in the processes of land-use changes and NPS pollution output can be effectively addressed. The developed model was applied to a water source area in the central part of northern Guangdong Province in South China. For the prediction period 2020–2030 under the high-speed development scenario, results show that cropland area would decrease, while grassland and waterbody areas would increase. In contrast, these three types of land-use would show opposite variation trends under the low-speed development scenario. Construction land area would decrease, while forestland area would increase under both the low-speed and high-speed development scenarios. Variation of the predicted land-use patterns would lead to an increase of total nitrogen loads under each of the scenario, while the total phosphorus loads would show relatively complex variation trends. Regional land-use patterns should be further optimized to mitigate NPS pollution. However, the pollution loads in the study area cannot be reduced by >5% through land-use adjustment. Because cropland would still be the critical source of NPS pollution after optimization, strictly controlling the areas of cropland would be important for the management of such pollution in the research area. In addition, certain areas of grassland and waterbody would need to be converted into cropland and construction land to balance the economic benefit of the system and NPS pollution control. Multiple results obtained from the model under different scenarios of pollution reduction targets and α-cut levels can provide decision-making supports for the local policy makers. The developed ILUPO model can yield insights useful for the planning and adjustment of regional land-use patterns while considering NPS pollution control under conditions of uncertainty.

A stochastic modeling approach for analyzing water resources systems

Many uncertain factors exist in the water resource systems, leading to dynamic characteristics of the water distribution process. Especially for the watershed including irrigation area with multiple water sources and water users, it is complicated that the joint risk among available water from each water source and multi-uncertainties under water resource allocation among multiple water users. In this research, an approach is developed that a copula-based interval multi-stage fuzzy stochastic programming (CIMFSP) model. This research aims at figuring out the planning of a case study for water-resources management. In the multi-sources water supply subsystem, the copula function is introduced to tackle the interaction of water availability between two water sources (i.e., A &B). Joint risk was set 0.05, 0.10, 0.15 to reflect the shortage risk of total available water at diverse levels. An interval parameter multi-stage fuzzy stochastic programming (IMFSP) model is developed for water resource distribution in a multi-users water-demand subsystem. Through this model, uncertainties presented as interval numbers and probability distributions, as well as fuzzy sets, were handled. The dynamics of the entire water resource system can be reflected by multi-stage discrete trees. A series of solutions can be generated under multiple scenarios (i.e., joint risk and α-cut levels). The modeling results will produce a series of alternatives under a battery of scenarios and help the decision-makers get an insight into the tradeoff between the system economic benefit and financial penalties under the corresponding risk level. This approach is valuable for improving the feasibility of optimal results in the watershed with irrigation region water resource management.

Discharge estimation for trapezoidal open channels applying fuzzy transformation method to a flow equation

Abstract The aim of this paper is to implement fuzzy logic theory in order to estimate the discharge for open channels, which is a well-known physical problem affected by many factors. The problem can be solved by the Manning equation but the parameters present uncertainties as to their true-real values. Especially, the Manning n roughness coefficient, which is an empirically derived coefficient, presents quite a high variation for different substrates. With the help of fuzzy logic and utilizing a fuzzy transformation method, it is possible to include the uncertainties of the problem in the calculation process. In this case, it is feasible to estimate the discharge, putting more emphasis on different uncertainty rates of the Manning roughness coefficient, while the rest of the parameters remain with constant or zero uncertainty level. By taking different α-cut levels, it is shown that the methodology gives realistic and reliable results, presenting with great accuracy the variations of the water discharge for trapezoidal open channels. In this way, a possible underestimation or overestimation of the actual physical condition is avoided, by helping engineers and researchers to obtain a more comprehensive view of the real physical conditions, and thus make better management plans.

Two-stage common weight DEA-Based approach for performance evaluation with imprecise data

A multi-criteria decision making approach based on data envelopment analysis (DEA) is presented to identify the best performing decision making unit (DMU) accounting for multiple inputs and multiple outputs with the presence of imprecise data. The developed α-cut based two-stage mathematical programming approach, which yields feasible solutions for all α-cut levels, generates common set of weights for inputs and outputs, and thus, provides more practical and realistic performance assessment of DMUs. A single rank-order is obtained through OWA operator that is employed for aggregating the efficiency scores regarding α-levels for each DMU. The robustness of the developed methodology is illustrated by examples taken from earlier research studies along with a case study that is conducted to aid an expatriate to identify the most desirable country in terms of quality of living. The proposed approach provides a ranking with improved discriminating power and enhanced weight dispersion with regard to inputs and outputs while also guaranteeing to determine a single best performing DMU.

Lean holistic fuzzy methodology employing cross-functional worker teams for new product development projects: A real case study from high-tech industry

This paper addresses a lean holistic fuzzy methodology for the new product development (NPD) projects to cope with the uncertainties encountered in the projects by placing an emphasis on the cross-functional worker teams along with utility workers’ impact on the lead time and total operational cost. To this end, the fuzzy design structure matrix (FDSM), the fuzzy value stream mapping (FVSM), and a novel fuzzy optimization model are combined and employed sequentially for visualizing all processes, decreasing lead time and operational cost, and determining the improvement points. While the whole methodology focuses on reducing the lead time by creating the worker teams, the fuzzy optimization model aims to decrease the total operational costs. The study is conducted in a startup, which has begun to reorganize its NPD projects according to lean principles by implementing the proposed methodology. The effectiveness of the methodology to reduce the lead time and the operational cost is shown in the results based on the data taken from the startup. To analyze the impact of parameters on the total operational cost in terms of the total number of workers (w.r.t. normal and utility) and the lead time as well, the computational experiments are carried out through a set of parameter values and α-cut levels. According to the results, the proposed methodology leads to a decrease in both the lead time and the total operational cost thanks to employing the utility worker concept.

Multi-objective decision-making for green infrastructure planning (LID-BMPs) in urban storm water management under uncertainty

Sustainable urban surface runoff management is receiving increased attention due to environmental and ecological consequences related to urbanization. This study presents a useful new framework for green infrastructure (GI) planning for the management of urban runoff quality and quantity. The proposed framework considers physical, technical, economic, and multi-stakeholder aspects related to urban runoff management while simultaneously addressing the uncertainties of the decision-making process and model parameters. The methodology is applied to regionally locating and sizing low impact development-best management practices (LID-BMPs) while considering the hydrologic and hydraulic impacts. A decision-making multi-objective optimization framework is developed by integrating: 1) a Multi-Layer Perceptron neural network founded on a Storm Water Management Model (SWMM-MLP meta-model), 2) NSGA-II multi-objective optimization, 3) fuzzy α-cut technique, and 4) a decision-making support model based on social choice theory to elicit trade-offs among system cost and LID-BMP performance indicators. The decision-making model, based on Fuzzy Social Choice (FSC) theory, is applied to simulate consensus between stakeholders for a partially cooperative group decision-making problem. The proposed methodology is explored in a catchment located in the northeastern part of Tehran, Iran. Results showed that the SWMM could be effectively replaced with a MLP-based meta-model in simulation-optimization problems for urban runoff management. In the application of FSC methods, the optimal scenarios were effective in reducing the volume of urban runoff and contamination loads while maintaining the optimality of the operation costs. Considering the optimal LID scenario, a reduction of more than 99% in runoff volume and biochemical oxygen demand (BOD), and a decrease of more than 92% of total suspended solids (TSS), occurred for the lower bound of uncertainty (lower (left) end of the α-cut = 0.3). For the upper bound of uncertainty (upper (right) end of the α-cut level = 0.3), a maximum reduction of 57% was obtained. In applying FSC methods through the decision-making process, the Borda Counting method considered the preferences of all stakeholders best. Moreover, the proposed framework allows decision-makers to decide on the acceptability and reliability of the optimal management scenarios considering their preferences and uncertainties.

An inexact fractional programming model for irrigation water resources optimal allocation under multiple uncertainties.

In reality, severe water shortage crisis has made bad impact on the sustainable development of a region. In addition, uncertainties are inevitable in the irrigation system. Therefore, a fully fuzzy fractional programming model for optimization allocation of irrigation water resources, which aimed at not only irrigation water optimization but also improving water use efficiency. And then the developed model applied to a case study in Minqin County, Gansu Province, China, which selected maximum economic benefit of per unit water resources as planning objective. Moreover, surface and underground water are main water sources for irrigation. Thus, conjunctive use of surface and underground water was taken under consideration in this study. By solving the developed model, a series of optimal crop area and planting schemes, which were under different α-cut levels, were offered to the decision makers. The obtained results could be helpful for decision makers to make decision on the optimal use of irrigation water resources under multiple uncertainties.

A fuzzy multi-stakeholder socio-optimal model for water and waste load allocation.

This study proposes a fuzzy multi-stakeholder socio-optimal methodology for joint water and waste load allocation (WWLA) in river systems while addressing upstream flow uncertainty and different social choice rules (SCRs). QUAL2Kw, as the numerical river water quality model, is executed for various scenarios of water and waste loads to construct a comprehensive dataset of plausible settings, which is in turn used to train a meta-model in the form of multivariate linear regressions. The river upstream flow as the main uncertain parameter is assessed by fuzzy transformation method (FTM). Then, for different confidence levels of fuzzy uncertain input, the meta-model is linked with the non-dominated sorting genetic algorithm (NSGA-II) multi-objective optimization model to generate trade-off curves among the stakeholders' utility functions. Subsequently, five SCRs are utilized at each confidence level to determine the fuzzy interval solutions for each objective. Next, the possibility degree method is applied to rank the fuzzy interval solutions in each α-cut level. Finally, considering the priorities of all stakeholders, the fallback bargaining method is used to specify the most appropriate SCR in each confidence level. Application of the proposed methodology in Kor River, Iran, shows its efficacy to realize the socio-optimal WWLA scenario(s) among different stakeholders.

Fuzzy Multi-Objective Simulation-Optimization of Stepped Spillways Considering Flood Uncertainty

The design of stepped spillways is a multi-objective optimization problem in which the uncertainty of parameters plays a key role in the stepped spillways’ design. A Fuzzy Transformation Method (FTM), as a first attempt, was employed to address the uncertainty through a multi-objective optimization of the stepped spillway. The proposed methodology comprises two parts. A numerical model is developed by FLOW3D® and the Multi-Layer Perceptron artificial neural network (MLP-ANN) meta-model was trained and validated to substitute the numerical model. Then, the meta-model is coupled with a multi-objective optimization model to find the optimal design scenarios in each α-cut level. Finally, two multi-criteria decision-making (MCDM) models are applied to achieve the lower and upper bounds of optimum solutions. The application of proposed methodology in Jarreh dam, Iran, shows its capability in satisfying the objectives and also addressing the uncertainty in stepped spillways’ design.

Fuzzy-based conflict resolution management of groundwater in-situ bioremediation under hydrogeological uncertainty

In this research study, a fuzzy multi-objective optimization methodology is proposed for in-situ groundwater bioremediation utilizing the Graph Model for Conflict Resolution. In the current model, uncertainties in hydraulic conductivity and the ratio of transverse to longitudinal dispersivity of contaminants are considered using the Fuzzy Transformation Method (FTM). First, the BIOPLUME III simulation model is linked with a Non-dominated Sorting Genetic Algorithm II (NSGA-II) multi-objective optimization model to optimize a groundwater bioremediation system regarding conflicting viewpoints of decision makers. Then, the hydrogeological uncertainties of the groundwater bioremediation system are included in the proposed methodology using FTM. The three main objectives of the in-situ bioremediation optimization model are overall cost (well installation, treatment or pumping, and facility capital), the sum of contaminant concentration violating any standards, and contaminant plume fragmentation, which need to be minimized based on stakeholders’ preferences. Subsequently, GMCR II is utilized to resolve any conflicts between the perspectives of the stakeholders to achieve a compromise solution. The performance assessment results represent the ability of the proposed methodology for optimal in-situ groundwater bioremediation. Results show that the minimum fuzzy interval is 32.5%, and pertains to the overall cost of the bioremediation system in fuzzy α-cut levels of 0 and 0.5. Conversely, the maximum fuzzy interval is related to contaminant plume fragmentation in fuzzy α-cut levels of 0 and 0.5, and was found to be 95.1%.

Developing a fuzzy optimization model for groundwater risk assessment based on improved DRASTIC method

Groundwater pollution is a serious threat to water resources which attracts hydrologists attention for sustainable management. Comprehensive assessment of groundwater vulnerability, however, requires uncertainty analysis in conjunction with a proper method to determine prone areas to contaminants. This paper provides a new fuzzy optimization methodology using improved DRASTIC method to model groundwater vulnerability risk assessment which considers uncertainties embedded in the input parameters, optimizes the weights and modifies the rates of the model simultaneously. To better represent hydrogeological characteristics of an area, the rating scores of original DRASTIC method is modified using Wilcoxon test considering nitrate concentration (as the main interfering pollutant in the study area). Spearman correlation coefficient between vulnerability indices and nitrate concentration is used as a factor to measure how well improved DRASTIC performs for vulnerability assessment as compared to the original method. The results show that the correlation coefficient significantly increased from 0.573 to 0.789. To address uncertainties associated with the input and output of the model, reduced fuzzy transformation method (FTM) is empowered by genetic algorithm (GA) to consider uncertainties associated with input parameters as well as optimizing weights of improved DRASTIC model. The results show how correlation coefficient changes at different uncertainty levels. Considering uncertainties in the inputs, correlation coefficient changes from 0.746 to almost 0.758 at α-cut level equal to 0 in comparison with that of equal to 1. Comparison of the risk maps of improved DRASTIC and fuzzy model at different uncertainty levels reveals that the model performs robustly under uncertainties mainly since the vulnerability trend and more importantly the severity of vulnerability indices have not changed remarkably. Based on these maps, east and southeastern parts of the study area are highly susceptible to contamination where intense industrial and agricultural activities can be seen. This framework provides helpful information for decision-makers to consider risk assessment at different uncertainty levels as it offers a continuous range of vulnerability indices rather than fixed ones. Also, vulnerability risk assessment maps demonstrate vulnerability trend throughout the area for further controlling or remedying actions of groundwater network.

Examination of differences in health indicators between efficient and inefficient countries.

Objective:This study determined whether there is a statistically significant difference between efficient and inefficient Organization for Economic Co-operation and Development (OECD) countries in terms of health indicators using fuzzy data envelopment analysis (FDEA).Methods:In the study, FDEA was performed with three input variables directly affecting health, four environmental factors considered to indirectly affect health, and two output variables. Literature research was used to determine appropriate variables. In FDEA, three different α-cut levels were used. The hypotheses regarding whether there was a statistically significant difference between efficient and inefficient countries in input and output variables were tested for all α-cut levels of upper bound efficiency values.Results:In terms of health indicators, 17 countries were efficient at α-cut 0 and 0.5. At α-cut 1, 18 countries were efficient. There was only a statistically significant difference between the efficient and inefficient countries in “the number of physicians.”Conclusion:This study shows the number of physicians was the most important determinant affecting the efficiency of a country’s healthcare system. Inefficient countries had a greater mean for number of physicians. Thus, inefficient countries consume more resources than efficient ones.

A multi-stage fuzzy stochastic programming method for water resources management with the consideration of ecological water demand

In this paper, a multi-stage fuzzy stochastic programming (MFSP) method is introduced to deal with uncertainties presented as fuzzy sets and probability distributions. Moreover, it is able to reflect dynamics of uncertainties and the related decision processes through constructing a series of representative scenarios within a multi-stage context under a set of fuzzy α-cut levels. A management problem about long-term planning of water resources system has been studied to illustrate applicability of the proposed approach. With ecological water demand being considered, the framework solves the complex problems that can hardly be solved in previous individual model research and promotes sustainable development. The results indicate that the dynamic and complexity of water resources allocation can be reflected through the multilayer discrete context tree. Moreover, real-time correction for reducing the risk of water shortage and low economic penalty can be presented. They can also help identify satisfaction degree of the goal and feasibility degree of constraints in an interactive way, enabling decision makers to generate a series of alternatives under various system conditions. Overall, it can not only contribute to decision makers for in-depth analysis, but also for sustainable development of ecosystem.

Interval Double-Sided Fuzzy Chance-Constrained Programming Model for Water Resources Allocation

Abstract In this study, an interval double-sided fuzzy chance-constrained programming (IDFCP) approach is developed for identifying water resources allocation strategies under uncertainty. Through incorporating interval parameter programming, double-sided fuzzy programming, and chance-constrained programming into a general framework, IDFCP can effectively deal with uncertainties expressed as intervals, probability distributions, and fuzzy sets. IDFCP can also examine the risk of violating system constraints. IDFCP is then applied to water resources allocation in the middle and upper reaches of Fen River Basin that is associated with multiuser, multiregion, and multisource features. Interval solutions of the compromise decision alternatives are generated under different scenarios in association with different risk levels of violating constraints (i.e., p levels), fuzzy membership degrees (i.e., α-cut levels), credible degrees (i.e., minimum and maximum), and reclaimed water utilization ratios. Results obta...

A fuzzy α-cut optimization analysis for vibration control of laminated composite smart structures under uncertainties

As a high performance and specific mechanically tailored material, composite structures present high structural behavior sensitivity to small variations in geometry or material properties. Some of these uncertainties are intrinsically related to the manufacturing process but others are usual uncertainty in material properties. A fuzzy interval analysis methodology is proposed in this paper in order to evaluate the structural behavior of laminated composite smart structures under vibration control and uncertain parameters. A Particle Swarm Optimization algorithm is used to search for input parameters in order to minimize and maximize system outputs by an anti-optimization and thus build their envelopes. Each input parameter has interval values defined by α-cut levels and interpolated by the Fuzzy reasoning. This methodology is applied to a smart structure of laminated composite material with attached piezoelectric patches and controlled by an optimal control regulator. System's interval outputs like natural frequency, mechanical vibration, and electric control input energy are investigated taking into account uncertainties in the composite and piezoelectric material properties, ply angles and layer thickness. Finally, it is concluded that these uncertainties may affect the control performance since it was found significant modifications in the dynamic behavior and therefore this should be accounted in the design stages.

A unified approach for squeal instability analysis of disc brakes with two types of random-fuzzy uncertainties

Automotive brake systems are always subjected to various types of uncertainties and two types of random-fuzzy uncertainties may exist in the brakes. In this paper, a unified approach is proposed for squeal instability analysis of disc brakes with two types of random-fuzzy uncertainties. In the proposed approach, two uncertainty analysis models with mixed variables are introduced to model the random-fuzzy uncertainties. The first one is the random and fuzzy model, in which random variables and fuzzy variables exist simultaneously and independently. The second one is the fuzzy random model, in which uncertain parameters are all treated as random variables while their distribution parameters are expressed as fuzzy numbers. Firstly, the fuzziness is discretized by using α-cut technique and the two uncertainty analysis models are simplified into random-interval models. Afterwards, by temporarily neglecting interval uncertainties, the random-interval models are degraded into random models, in which the expectations, variances, reliability indexes and reliability probabilities of system stability functions are calculated. And then, by reconsidering the interval uncertainties, the bounds of the expectations, variances, reliability indexes and reliability probabilities are computed based on Taylor series expansion. Finally, by recomposing the analysis results at each α-cut level, the fuzzy reliability indexes and probabilities can be obtained, by which the brake squeal instability can be evaluated. The proposed approach gives a general framework to deal with both types of random-fuzzy uncertainties that may exist in the brakes and its effectiveness is demonstrated by numerical examples. It will be a valuable supplement to the systematic study of brake squeal considering uncertainty.