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We propose a new concept which is a generalization of fuzzy soft subset and fuzzy soft equal. Using such notions, we will be able to consider the distributive law of fuzzy soft sets. Using the distributive law of fuzzy soft sets, we point out that the distributive law of trapezoidal fuzzy soft sets as proposed by Xiao et al. (2012) is not true. The correction will further improve further extensions of the results of Xiao et al. (2012). We will also establish the generalized distributive law of trapezoidal fuzzy soft sets along with illustrative examples. [ABSTRACT FROM AUTHOR]

• Proposed fuzzy small displacement torsor model for tolerance classification representation. • Established an assembly reliability index for evaluating assembly precision. • Developed constrained transformation method for efficient and high-accuracy calculation. • Proposed a novel fuzzy algorithm for assembly precision management. The assembly precision of a specified product is subject to epistemic uncertainty due to manufacturing and measurement errors. The efficient and accurate assembly precision management is essential for realizing smart production lines, which rely on the robust assembly precision analysis model and calculation method. Parametric models are extensively used for assembly precision analysis in industrial products, typically integrating worst-case and statistical methods for calculations. Nevertheless, the combined application of these methods presents inherent limitations. Therefore, a novel fuzzy algorithm for assembly precision management is proposed in this paper by using fuzzy sets to quantify epistemic uncertainty in assembly precision, which integrates the proposed fuzzy-based assembly precision analysis model and fuzzy-based calculation method. The proposed fuzzy small displacement torsor model is a fuzzy-based model for deviation representation and tolerance classification using decomposed fuzzy numbers, where an assembly reliability index is always established for the hierarchical evaluation and management of assembly precision. Subsequently, a comprehensive assembly precision analysis model is developed for precision prediction and contribution quantification by integrating the Jacobian model for deviation propagation. A new constrained transformation method is proposed as a fuzzy-based calculation method, offering efficient and highly accurate assembly precision computation. It accounts for torsor parameter constraints to ensure greater prediction accuracy than the worst-case method and improves computational efficiency compared to statistical methods. An assembly case of the centering pin mechanism is applied to verify the superiority of the proposed novel fuzzy algorithm compared with the Jacobian-Torsor model with worst-case or statistical methods. [ABSTRACT FROM AUTHOR]

• A phase duration calculation method is proposed based on multi-phase identification. • The components' fuzzy state probability vectors are acquired using the evidential reasoning rule. • A performance evaluation model of cyclic phased-mission systems is constructed by multi-valued decision diagrams. • A phase duration adjustment method based on the evaluation results is proposed. • The proposed method is verified by a performance evaluation case study on satellite turntable system. Phased-mission systems (PMS) often perform different missions that have multiple consecutive, non-overlapping phases. Many recent studies have focused on the reliability analysis for PMS. However, not much attention has been paid to the PMS performance evaluation and the phase cycle characteristic of PMS. In this paper, a performance evaluation method of non-repairable cyclic phased-mission systems (CPMS) is proposed. Considering the uncertainty caused by the change of phase duration and component performance degradation, the average duration of each phase and the fuzzy state probability vector of each component are obtained by the evidential reasoning rule (ER rule). Then, the multi-valued decision diagrams (MDD) manipulation rules suitable for cyclic phase conditions are proposed, and the component state composition path generation method based on cycle-end component states is developed. On the above basis, MDD models for CPMS (CPMS-MDD) in different states are established, and the system performance is evaluated by the continuous time Markov chains (CTMC). A phase duration adjustment strategy is further proposed based on the evaluation results. The effectiveness and correctness of the proposed method are verified by case studies on performance evaluation of satellite turntable system. Comparative study shows the concise and efficient of the proposed method. [ABSTRACT FROM AUTHOR]

• Multidisciplinary design optimization of turbine twin-web disk using hybrid uncertain model. • Non-probabilistic reliability assessment by modified evidence theory based fuzzy set model. • Suitable for both general and Bayesian data structure in evidence theory. • Reducing computational cost by multi-adaptive active learning kriging. The twin-web disk holds big promise for increasing efficiency of the aircraft engine. Its reliability-based multidisciplinary design optimization involves several disciplines including fluid mechanics, heat transfer, structural strength, and vibration. The solution to this optimization problem requires three-loop calculations including loops for optimization, reliability, and interdisciplinary consistence often making its computational cost unacceptably high. The lack of sufficient amount of probabilistic data, especially for this brand-new turbine disk, makes matters worse. In this paper, the non-probabilistic uncertain variables are described by an evidence theory-based fuzzy set method, which we extend to general structure of uncertain data. We also propose two modifications of the active learning kriging model: one of them for the purpose of optimization with respect to the distance from the optimum point and another one for the purpose of assessing reliability by introducing the importance concept. Applications of these two modifications are demonstrated in this paper. Finally, a multi-adaptive learning kriging strategy for non-probabilistic reliability-based multidisciplinary design optimization of twin-web disk is proposed to improve its power efficiency and reliability in a computationally effective way. [ABSTRACT FROM AUTHOR]

• Comparison of experimental and numerical buckling variability of a set of additively manufactured lattice structures. • Reduction of computational time for prediction, sensitivity and optimization steps of an uncertain propagation problem. • Efficient coupling between a Reduced Order Model and a surrogate model. • Improvement of the calculation precision with projection basis composed of high order perturbed modes. This paper presents a new method to efficiently approximate both linear buckling loads and associated mode shapes of finite element structures subject to perturbations. To achieve this, a coupling between a Reduced Order Model (ROM) based on the Homotopy Perturbation Method (HPM) and a Kriging model is presented here. The ROM maintains the link between eigenvalues, related eigenvectors and the dependencies between each eigenvector components, leading to a high precision level. The computational time is greatly reduced by the surrogate model which avoids the computation of modified finite element matrices for each prediction. Next, the capabilities of the method allow to efficiently handle the prediction, sensitivity and optimization steps of an uncertain propagation problem using fuzzy formalism. Additive Manufacturing is a powerful and impressive process but many factors can be responsible for relatively large discrepancies in the mechanical and geometrical characteristics of the manufactured structure. Lastly, a study shows how the proposed fuzzy strategy allows the prediction of the buckling variability of a set of lattice structures. [ABSTRACT FROM AUTHOR]

• Solutions to vibration problems with interactive fuzzy numbers as initial conditions. • These are obtained by extending deterministic solutions via Zadeh and sup-J extension. • Examples of coupled/uncoupled vibration problems with fuzzy uncertainties are studied. • Interactive fuzzy solution represents in a better way the system's physical behavior. Fuzzy initial value problems describing classical mechanical vibrations are the focus of this paper. In particular, this work considers systems described by n th-order linear ordinary differential equations whose initial conditions are uncertain and given by interactive fuzzy numbers. The concept of interactivity arises from the concept of joint possibility distribution (J). An approach based on the sup- J extension principle, which is a generalization of Zadeh's extension principle, is presented. This theory is applied to two major examples of oscillatory systems: the forced vibration of an uncoupled mass-spring-damper system and the free vibration of a coupled undamped mass-spring system. In both cases, we have that the solution via sup- J extension, where the fuzzy initial conditions are given by linearly correlated fuzzy numbers, is contained in the solution via Zadeh's extension. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a multiple attribute group decision making model based on aggregating crisp values into intuitionistic fuzzy numbers. First, each alternative is evaluated with respect to their attributes, whose values are provided by decision maker as crisp numbers. Second, to make a reasonable normalization of attribute values in the group decision making environment, a maximum grade and a minimum grade are added to the attribute values. These normalized attribute values are then aggregated (per attribute) into an induced intuitionistic fuzzy number. Each alternative is then evaluated according to the induced intuitionistic fuzzy number. To show the major technical advances in this paper, comparisons with other methods are also made. Finally, an experimental analysis for supplier selection is given to illustrate the reasonableness and efficiency of the introduced method. [Copyright &y& Elsevier]

Abstract: Although the rough set and intuitionistic fuzzy set both capture the same notion, imprecision, studies on the combination of these two theories are rare. Rule extraction is an important task in a type of decision systems where condition attributes are taken as intuitionistic fuzzy values and those of decision attribute are crisp ones. To address this issue, this paper makes a contribution of the following aspects. First, a ranking method is introduced to construct the neighborhood of every object that is determined by intuitionistic fuzzy values of condition attributes. Moreover, an original notion, dominance intuitionistic fuzzy decision tables (DIFDT), is proposed in this paper. Second, a lower/upper approximation set of an object and crisp classes that are confirmed by decision attributes is ascertained by comparing the relation between them. Third, making use of the discernibility matrix and discernibility function, a lower and upper approximation reduction and rule extraction algorithm is devised to acquire knowledge from existing dominance intuitionistic fuzzy decision tables. Finally, the presented model and algorithms are applied to audit risk judgment on information system security auditing risk judgement for CISA, candidate global supplier selection in a manufacturing company, and cars classification. [Copyright &y& Elsevier]

Abstract: Decision Making Trial and Evaluation Laboratory (DEMATEL) has been applied in many situations, such as marketing strategies, control systems, safety problems, developing the competencies of global managers and group decision making. It has been incorporated into other methods such as Analytical Network Process (ANP), Multiple Criteria Decision Making (MCDM), fuzzy set theory, etc., to vitalize these traditional methods and explore new applications for the hybrid methods. DEMATEL models the influences of components of a system with an initial direct relation matrix. Influences of components can ripple transitively to other components, which is modeled by raising the initial direct relation matrix to powers. The total influence is computed by summing up matrices of all powers based on the assumption that the matrix raising to the power of infinity would converge to zero. The current paper shows that raising the initial relation matrix to the power of infinity may not converge to zero and hence total influence may not converge. The current paper also shows that our revised DEMATEL guarantees that the initial direct-relation matrix to infinite power will converge to zero and the total influence can be obtained accordingly. The newly developed approach is illustrated with numerical examples. [Copyright &y& Elsevier]

Abstract: This paper points out three questionable areas in the realm of similarity measures and then provides a new method that will rectify the problem. The purpose of this paper is fourfold. First, we will propose a scenario where the three similarity measures proposed by Hung and Yang (2004) [1] are helpless in aiding a decision maker in deciding pattern recognition problem. Second, we will present our method for solving the dilemma. Third, we will show that our proposed similarity measures satisfy the axioms for well defined similarity measures. Fourth, we will prove that our method could solve pattern recognition problems. Our findings will help researchers handle similarity problems under intuitionistic fuzzy sets environment. [Copyright &y& Elsevier]

Abstract: Soft set theory is a newly emerging mathematical tool to deal with uncertain problems. Since the trapezoidal fuzzy number, as a vital concept of fuzzy set, can express linguistic assessments by transforming them into numerical variables objectively, this paper aims to extend classical soft sets to trapezoidal fuzzy soft sets based on trapezoidal fuzzy numbers. Then, some operations on a trapezoidal fuzzy soft set are defined, such as complement operation, “AND” operation, and “OR” operation. Finally, a Multiple Criterion Decision-Making (MCDM) problem under a fuzzy environment is analyzed by trapezoidal fuzzy soft sets with the demonstration of a numerical example. This paper also uses traditional fuzzy soft sets to deal with the MCDM problem. The result shows that the method provided by this paper outperforms the traditional one. [Copyright &y& Elsevier]

Abstract: Soft set theory, initiated by Molodtsov, is a general mathematical tool for dealing with uncertain problems. In this paper, we first point out that the similarity measure in a previous paper by Majumdar and Samanta [P. Majumdar, S.K. Samanta, Generalized fuzzy soft sets, Comput. Math. Appl. 59 (2010) 1425–1432] is limited by two counterexamples. To deal with the problems of subjective evaluation and uncertain knowledge, this paper proposes the concept of D–S generalized fuzzy soft sets by combining Dempster–Shafer theory of evidence and generalized fuzzy soft sets. We study some of its operations and basic properties, and the relationship between generalized fuzzy soft sets and D–S generalized fuzzy soft sets are introduced. Then we propose the concept of the similarity between two D–S generalized fuzzy soft sets. At last, we present a new method of evaluation based on D–S generalized fuzzy soft sets and apply it into a medical diagnosis problem. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to present a novel fuzzy modified technique of order preference by a similarity to ideal solution (TOPSIS) method by a group of experts, which can select the best alternative by considering both conflicting quantitative and qualitative evaluation criteria in real-life applications. The proposed method satisfies the condition of being the closest to the fuzzy positive ideal solution and also being the farthest from the fuzzy negative ideal solution with multi-judges and multi-criteria. The performance rating values of alternatives versus conflicting criteria as well as the weights of criteria are described by linguistic variables and are transformed into triangular fuzzy numbers. Then a new collective index is introduced to discriminate among alternatives in the evaluation process with respect to subjective judgment and objective information. This paper shows that the proposed fuzzy modified TOPSIS method is a suitable decision making tool for the manufacturing decisions with two examples for the robot selection and rapid prototyping process selection. [Copyright &y& Elsevier]

Abstract: The concept of the rough set was originally proposed by Pawlak as a formal tool for modeling and processing incomplete information in information systems. Then in 1990, Dubois and Prade first introduced the rough fuzzy sets and fuzzy rough sets as a fuzzy extension of the rough sets. The aim of this paper is to present a new extension of the rough set model on the different universe. i.e., the fuzzy rough sets model between two different universes is presented based on the fuzzy compatible relation , which is defined by a fuzzy relation between two different non-empty universes U and V and the threshold α(α ∈(0,1]). Several properties of this rough sets model are given, and the relationships of this model with others rough set model are examined, too. Furthermore, we also discuss two extended models of the fuzzy rough sets model based on the fuzzy rough set model on different universes. i.e., the degree fuzzy rough set and the variable precision fuzzy rough set. Meanwhile, some principal conclusions of this two rough sets model are also established. Finally, a test example is applied to interpret the background of the application and the ideals for the fuzzy rough sets model that is presented in this paper. [Copyright &y& Elsevier]

Abstract: This paper proposes a decomposition method for hierarchical generation of α-Pareto optimal solutions in large-scale multi-objective non-linear programming (MONLP) problems with fuzzy parameters in the objective functions and in the constraints (FMONLP). These fuzzy parameters are characterized by fuzzy numbers. For such problems, the concept of α-Pareto optimality introduced by extending the ordinary Pareto optimality based on the α-level sets of fuzzy numbers. The decomposition method is based on the principle of decompose the original problem into interdependent sub-problems. In this method, the global multi-objective non-linear problem is decomposed into smaller multi-objective sub-problems. The smaller sub-problems, which obtained solved separately by using the weighting method and through an operative procedure. All these solution are coordinates in such a way that an optimal solution for the global problem achieved. In addition, an interactive fuzzy decision-making algorithm for hierarchical generation of α-Pareto optimal solution through the decomposition method is developed. Finally, two numerical examples given to illustrate the results developed in this paper. [Copyright &y& Elsevier]

In this paper, the Takagi–Sugeno fuzzy model is derived from the given nonlinear systems. The objective is to linearize these nonlinear systems into several fuzzy differential equations according to the Takagi–Sugeno fuzzy rules. The present work implemented the nontraditional ant colony programming (ACP) method to solve these fuzzy differential equations. The proposed ACP algorithm manages to give either similar or almost close solutions to the analytical form. Accuracy of the solution computed by this ACP method is qualitatively better when it is compared with other nontraditional approaches such as the genetic programming (GP) method. Illustrative numerical examples and tables are presented for comparative purpose. [ABSTRACT FROM AUTHOR]

This paper deals with the fuzzy solid transportation problem (FSTP) that has fuzzy cost coefficients, fuzzy supplies, fuzzy demands and fuzzy conveyances. All these fuzzy quantities of FSTP are assumed to be triangular fuzzy numbers. For this problem, we propose an approach to generate all optimal solutions parametrically. The first stage of our approach is to determine the feasibility range based on fuzzy supply–demand–conveyance quantities. In the second stage, the breaking points of fuzzy costs are found by intersecting the membership functions of the fuzzy costs. The last stage constructs the optimal solutions of FSTP by means of some proposed auxiliary programs. Also a numerical example has been provided to illustrate our solution procedure. [ABSTRACT FROM AUTHOR]

Correlation between two variables or parameters plays a very significant role in statistics. Furthermore, the accuracy in the measurement of the correlation depends upon the data collected for the set of discourse. It is quite evident that in many cases the data collected for various statistical measures are full of uncertainties. A dual hesitant fuzzy set (DHFS) is a generalized form of a hesitant fuzzy set (HFS) and negates the effects of uncertainty inherent in the collected data. In the present paper, the concept of HFS has been replaced with DHFS and the correlation between two DHFSs is obtained. A formula for the correlation coefficient between two DHFSs has been derived. The proposed method was used to determine the coefficient of correlation between different parameters of water in four different lakes in Rajasthan, India. [ABSTRACT FROM AUTHOR]

The aim of this paper is to improve the applicability of the fuzzy-AHP method by using the rotational priority investigation (RPI) method. Despite its popularity and convenience, the AHP and fuzzy-AHP method is criticized by many scholars because of intransitivity and the rank reversal phenomenon. Experts may question the rational choice theory and cross priorities may indicate conflicting interactions. Also, the extraction of a number of alternatives may cause a different order of priorities. The rotational priority investigation method is proposed to investigate sub-group priorities and their corresponding rankings. Every rotation refers to the investigation of sub-group priorities after extraction of an alternative. The final result is based on the normalized average priority among the several rotations. An empirical study is presented by using the RPI method in the fuzzy AHP for the marine engine selection problem in the shipping industry. [ABSTRACT FROM AUTHOR]

Abstract: The theory of interval-valued intuitionistic fuzzy sets is useful and beneficial for handling uncertainty and imprecision in multiple criteria decision analysis. In addition, the theory allows for convenient quantification of the equivocal nature of human subjective assessments. In this paper, by extending the traditional linear assignment method, we propose a useful method for solving multiple criteria evaluation problems in the interval-valued intuitionistic fuzzy context. A ranking procedure consisting of score functions, accuracy functions, membership uncertainty indices, and hesitation uncertainty indices is presented to determine a criterion-wise preference of the alternatives. An extended linear assignment model is then constructed using a modified weighted-rank frequency matrix to determine the priority order of various alternatives. The feasibility and applicability of the proposed method are illustrated with a multiple criteria decision-making problem involving the selection of a bridge construction method. Additionally, a comparative analysis with other methods, including the approach with weighted aggregation operators, the closeness coefficient-based method, and the auxiliary nonlinear programming models, has been conducted for solving the investment company selection problem to validate the effectiveness of the extended linear assignment method. [Copyright &y& Elsevier]

Abstract: A single valued neutrosophic set (SVNS) is an instance of a neutrosophic set, which give us an additional possibility to represent uncertainty, imprecise, incomplete, and inconsistent information which exist in real world. It would be more suitable to apply indeterminate information and inconsistent information measures. In this paper, the cross entropy of SVNSs, called single valued neutrosophic cross entropy, is proposed as an extension of the cross entropy of fuzzy sets. Then, a multicriteria decision-making method based on the proposed single valued neutrosophic cross entropy is established in which criteria values for alternatives are SVNSs. In decision making process, we utilize the single-valued neutrosophic weighted cross entropy between the ideal alternative and an alternative to rank the alternatives corresponding to the cross entropy values and to select the most desirable one(s). Finally, a practical example of the choosing problem of suppliers is provided to illustrate the application of the developed approach. [Copyright &y& Elsevier]

Abstract: This paper investigates an approach for multi-criterion decision making (MCDM) problems with interval-valued intuitionistic fuzzy preference relations (IVIFPRs). Based on the novel interval score function, some extended concepts associated with IVIFPRs are defined, including the score matrix, the approximate optimal transfer matrix and the possibility degree matrix. By using these new matrixes, a prioritization method for IVIFPRs is proposed. Then, we investigate an interval-valued intuitionistic fuzzy AHP method for multi-criteria decision making (MCDM) problems. In the end, a numerical example is provided to illustrate the application of the proposed approach. [Copyright &y& Elsevier]

Abstract: In the real world of production, there are some uncertain parameters such as capacity, profit, and processing time. On the other hand, the fuzzy sets theory is applied as a suitable tool for managing production where dynamicity of the production environment prevents from determining target function, constraints and other parameters of the model. In this paper, a genetic algorithm embedded in fuzzy revised theory of constraints is presented for product mix problems. Results of implementing the proposed algorithm show efficiency and flexibility of this algorithm. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to solve a supplier selection problem under multi-price level and multi-product using interactive two-phase fuzzy multi-objective linear programming (FMOLP) model. The proposed model attempts to simultaneously minimize total purchasing and ordering costs, a number of defective units, and late delivered units ordered from suppliers. The piecewise linear membership functions are applied to represent the decision maker’s fuzzy goals for the supplier selection and order allocation problem, and can be resulted in more flexibility via an interactive decision-making process. To demonstrate effectiveness of the proposed model, results of applying the proposed model are shown by a numerical example. The analytical results show that the proposed approach is effective in uncertain environments and provide a reliable decision tool for integrated multi-objective supplier selection problems. [Copyright &y& Elsevier]

Abstract: Ordering fuzzy quantities and their comparison play a key tool in many applied models in the world and in particular decision-making procedures. However a huge number of researches is attracted to this filed but until now there is any unique accepted method to rank the fuzzy quantities. In fact, each proposed method may has some shortcoming. So we are going to present a novel method based on the angle of the reference functions to cover a wide range of fuzzy quantities by over coming the draw backs of some existing methods. In the mentioned firstly, the angle between the left and right membership functions (the reference functions) of every fuzzy set is called Angle of Fuzzy Set (AFS), and then in order to extend ranking of two fuzzy sets the angle of fuzzy sets and α-cuts is used. The method is illustrated by some numerical examples and in particular the results of ranking by the proposed method and some common and existing methods for ranking fuzzy sets is compared to verify the advantage of the new approach. In particular, based on the results of comparison of our method with well known methods which are exist in the literature, we will see that against of most existing ranking approaches, our proposed approach can rank fuzzy numbers that have the same mode and symmetric spreads. In fact, the proposed method in this paper can effectively rank symmetric fuzzy numbers as well as the effective methods which are appeared in the literature. Moreover, unlike of most existing ranking approaches, our proposed approach can rank non-normal fuzzy sets. Finally, we emphasize that the concept of fuzzy ordering is one of key role in establishing the numerical algorithms in operations research such as fuzzy primal simplex algorithms, fuzzy dual simplex algorithms and as well as discussed in the works of Ebrahimnejad and Nasseri and coworkers [1–7]. [Copyright &y& Elsevier]

Abstract: Molodtsov’s soft set theory was originally proposed as a general mathematical tool for dealing with uncertainty. By combining the multi-fuzzy set and soft set models, the purpose of this paper is to introduce the concept of multi-fuzzy soft sets. Some operations on a multi-fuzzy soft set are defined, such as complement operation, “AND” and “OR” operations, Union and Intersection operations. Then, the DeMorgan’s laws are proved. Finally, by means of level soft set, an algorithm is presented, and a decision problem is analyzed using multi-fuzzy soft set. [Copyright &y& Elsevier]

Abstract: Soft set theory, originally proposed by Molodtsov, has become an effective mathematical tool to deal with uncertainty. A type-2 fuzzy set, which is characterized by a fuzzy membership function, can provide us with more degrees of freedom to represent the uncertainty and the vagueness of the real world. Interval type-2 fuzzy sets are the most widely used type-2 fuzzy sets. In this paper, we first introduce the concept of trapezoidal interval type-2 fuzzy numbers and present some arithmetic operations between them. As a special case of interval type-2 fuzzy sets, trapezoidal interval type-2 fuzzy numbers can express linguistic assessments by transforming them into numerical variables objectively. Then, by combining trapezoidal interval type-2 fuzzy sets with soft sets, we propose the notion of trapezoidal interval type-2 fuzzy soft sets. Furthermore, some operations on trapezoidal interval type-2 fuzzy soft sets are defined and their properties are investigated. Finally, by using trapezoidal interval type-2 fuzzy soft sets, we propose a novel approach to multi attribute group decision making under interval type-2 fuzzy environment. A numerical example is given to illustrate the feasibility and effectiveness of the proposed method. [Copyright &y& Elsevier]

Abstract: The multiple criteria decision making (MCDM) methods VIKOR and TOPSIS are all based on an aggregating function representing “closeness to the ideal”, which originated in the compromise programming method. The VIKOR method of compromise ranking determines a compromise solution, providing a maximum “group utility” for the “majority” and a minimum of an “individual regret” for the “opponent”, which is an effective tool in multi-criteria decision making, particularly in a situation where the decision maker is not able, or does not know to express his/her preference at the beginning of system design. The TOPSIS method determines a solution with the shortest distance to the ideal solution and the greatest distance from the negative-ideal solution, but it does not consider the relative importance of these distances. And, the hesitant fuzzy set is a very useful tool to deal with uncertainty, which can be accurately and perfectly described in terms of the opinions of decision makers. In this paper, we develop the E-VIKOR method and TOPSIS method to solve the MCDM problems with hesitant fuzzy set information. Firstly, the hesitant fuzzy set information and corresponding concepts are described, and the basic essential of the VIKOR method is introduced. Then, the problem on multiple attribute decision marking is described, and the principles and steps of the proposed E-VIKOR method and TOPSIS method are presented. Finally, a numerical example illustrates an application of the E-VIKOR method, and the result by the TOPSIS method is compared. [Copyright &y& Elsevier]

Abstract: Incomplete data in soft sets lead to uncertainty and inaccuracy in representing and handling information. This paper introduces notions of complete distance between two objects and relative dominance degree between two parameters. Based on both the notions, an object-parameter method is proposed to predict unknown data in incomplete fuzzy soft sets. The proposal makes full use of known data, including the information from the relationship between known values of all objects on a certain parameter and the information from the relationship between known values of an object on all parameters. The effectiveness of the proposal is verified by many examples under the compared investigation of classical predicted methods. [Copyright &y& Elsevier]

Abstract: In the literature, it is pointed out that it is better to use vague sets instead of fuzzy sets. Several authors have proposed different methods for solving such differential equations in which all the parameters are represented by fuzzy numbers but to the best of our knowledge till now no one have represented the same as vague sets. In this paper, a new representation of (α, β)-cut, named as JMD (α, β)-cut, is proposed and with the help of JMD (α, β)-cut a new method is proposed to find the analytical solution of vague differential equations. To show the application of proposed method in real life problems the vague Kolmogorov’s differential equations, obtained by using vague Markov model of piston manufacturing system, are solved by proposed method. Also, to show the advantage of JMD (α, β)-cut over existing (α, β)-cut the same vague Kolmogorov’s differential equations are solved by using the proposed method with the help of existing (α, β)-cut and it is shown that the obtained results are not necessarily vague sets while the results, obtained by using JMD (α, β)-cut, are always vague sets. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to design a new extension of the ELECTRE, known as the elimination and choice translating reality method, for multi-criteria group decision-making problems based on intuitionistic fuzzy sets. This method is widely utilized when a set of alternatives should be identified and evaluated with respect to a set of conflicting criteria by reflecting decision makers’ (DMs’) preferences. However, handling the exact data and numerical measure is difficult to be precisely focused because the DMs’ judgments are often vague in real-life decision problems and applications. A more realistic and practical approach can be to use linguistic variables expressed in intuitionistic fuzzy numbers instead of numerical data to model DMs’ judgments and to describe the inputs in the ELECTRE method. The proposed intuitionsitic fuzzy ELECTRE utilizes the truth-membership function and non-truth-membership function to indicate the degrees of satisfiability and non-satisfiability of each alternative with respect to each criterion and the relative importance of each criterion, respectively. Then, a new discordance intuitionistic index is introduced, which is extended from the concept of the fuzzy distance measure. Outranking relations are defined by pairwise comparisons and a decision graph is depicted to determine which alternative is preferable, incomparable or indifferent in the intuitionistic fuzzy environment. Finally, a comprehensive sensitivity analysis is employed to further study regarding the impact of threshold values on the final evaluation, and a comparative analysis is demonstrated with an application example in flexible manufacturing systems between the proposed ELECTRE method and the existing intuitionistic fuzzy technique for order preference by similarity to ideal solution (IF-TOPSIS) method. [Copyright &y& Elsevier]

Abstract: The soft set theory, originally proposed by Molodtsov, can be used as a general mathematical tool for dealing with uncertainty. Since its appearance, there has been some progress concerning practical applications of soft set theory, especially the use of soft sets in decision making. The intuitionistic fuzzy soft set is a combination of an intuitionistic fuzzy set and a soft set. The rough set theory is a powerful tool for dealing with uncertainty, granuality and incompleteness of knowledge in information systems. Using rough set theory, this paper proposes a novel approach to intuitionistic fuzzy soft set based decision making problems. Firstly, by employing an intuitionistic fuzzy relation and a threshold value pair, we define a new rough set model and examine some fundamental properties of this rough set model. Then the concepts of approximate precision and rough degree are given and some basic properties are discussed. Furthermore, we investigate the relationship between intuitionistic fuzzy soft sets and intuitionistic fuzzy relations and present a rough set approach to intuitionistic fuzzy soft set based decision making. Finally, an illustrative example is employed to show the validity of this rough set approach in intuitionistic fuzzy soft set based decision making problems. [Copyright &y& Elsevier]

Abstract: This paper considers a construction project problem under multiple criteria in a fuzzy environment and proposes a new two-phase group decision making (GDM) approach. This approach integrates a modified analytic network process (ANP) and an improved compromise ranking method, known as VIKOR. To take uncertainty and risk into account, a new decision making approach is presented with multiple fuzzy information by a group of experts, and a risk attitude for each expert is incorporated that can be expressed linguistically. First, a modified fuzzy ANP method is introduced to address the problem of dependence as well as feedback among conflicting criteria and to determine their relative importance. Then, a fuzzy VIKOR method is extended to rank potential projects on the basis of their overall performance. An illustrative example from the literature is provided for the construction project problem to demonstrate the effectiveness and feasibility of the proposed approach. The computational results show that the proposed two-phase GDM approach is suitable to cope with imprecision and subjectivity for the complicated decision making problem. Finally, the associated results of the proposed approach with risk attitudes and without risk attitudes are compared with the results reported by Cheng and Li , and the merits are highlighted. [Copyright &y& Elsevier]

Abstract: This paper proposes the concept of the reduct intuitionistic fuzzy sets of interval-valued intuitionistic fuzzy sets (IVIFSs) with respect to adjustable weight vectors and the Dice similarity measure based on the reduct intuitionistic fuzzy sets to explore the effects of optimism, neutralism, and pessimism in decision making. Then a decision-making method with the pessimistic, optimistic, and neutral schemes desired by the decision maker is established by combining adjustable weight vectors and the Dice similarity measure for IVIFSs. The proposed decision-making method is more flexible and adjustable in practical problems and can determine the ranking order of alternatives and the optimal one(s), so that it can overcome the difficulty of the ranking order and decision making when there exist the same measure values of some alternatives in some cases. This adjustable feature can provide the decision maker with more selecting schemes and actionable results for the decision-making analysis. Finally, two illustrative examples are employed to show the feasibility of the proposed method in practical applications. [Copyright &y& Elsevier]

Abstract: Flood disasters are one of the most common and destructive natural hazards all over the world. In this paper, improved interior-outer-set model (IIOSM) based on information diffusion theory is introduced in detail to assess flood risk in an effort to obtain accurate analytical results that represent the actual situation. Then fuzzy α-cut technique is applied to calculate the fuzzy expected values under the possibility–probability distribution (PPD) calculated by IIOSM. Taking the value of α throughout the interval (0,1], we correspondingly get access to the conservative risk value (R C ) and venture risk value (R V ). Selection of α, R C and R V is dependent on present technical conditions and risk preference of different people. To illustrate the procedure of IIOSM and fuzzy α-cut technique, we employ them respectively to analyze the flood risk in Sanshui District, located in the center of Guangdong province in China. The results, such as risk value estimations, as well as fuzzy expected values, i.e. R C and R V under the given α-cut level, can reflect the flood risk quite accurately. The outcomes of this research based on IIOSM and fuzzy α-cut technique offer new insights to carry out an efficient way for various flood protection strategies. [Copyright &y& Elsevier]

Abstract: Decision-making information provided by decision makers is often imprecise or uncertain, due to lack of data, time pressure, or the decision makers’ limited attention and information-processing capabilities. Interval-valued fuzzy sets are associated with greater imprecision and more ambiguity than are ordinary fuzzy sets. For these reasons, this paper presents a signed distance-based method for handling fuzzy multiple-criteria group decision-making problems in which individual assessments are provided as generalized interval-valued trapezoidal fuzzy numbers, and the information about criterion weights are not precisely but partially known. First, concerning the relative importance of decision makers and the group consensus of fuzzy opinions, all individual decision opinions were aggregated into group opinions using a hybrid average with weighted averaging and signed distance-based ordered weighted averaging operations. Next, considering a decision situation with incomplete weight information of criteria, an integrated programming model was developed to estimate criterion weights and to order the priorities of various alternatives based on signed distances. In addition, several deviation variables were introduced to mitigate the effect of inconsistent evaluations on the importance of criteria. Finally, the feasibility of the proposed method is illustrated by a numerical example of a multi-criteria supplier selection problem. Furthermore, a comparative analysis with other methods was conducted to validate the effectiveness and applicability of the proposed methodology. [Copyright &y& Elsevier]

Abstract: The multiple attribute group decision making (MAGDM) problem with intuitionistic fuzzy information investigated in this paper is very useful for solving complicated decision problems under uncertain circumstances. Since experts have their own characteristics, they are familiar with some of the attributes, but not others, the weights of the decision makers to different attributes should be different. We derive the weights of the decision makers by aggregating the individual intuitionistic fuzzy decision matrices into a collective intuitionistic fuzzy decision matrix. The expert has a big weight if his evaluation value is close to the mean value and has a small weight if his evaluation value is far from the mean value. For the incomplete attribute weight information, we establish some optimization models to determine the attribute weights. Furthermore, we develop several algorithms for ranking alternatives under different situations, and then extend the developed models and algorithms to the MAGDM problem with interval-valued intuitionistic fuzzy information. Numerical results finally illustrate the practicality and efficiency of our new algorithms. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the problem of alternative-fuel buses selection using two novel fuzzy multiple criteria decision-making (MCDM) methods. Several types of fuels are considered as fuel modes, i.e., electricity, fuel cell (hydrogen), and methanol. For the purpose of determining appropriate fuel buses, many factors including quantitative and qualitative ones such as efficiency, price, and capability must be taken into account. In the first proposed method, linguistic variables are used to assess the ratings and weights for factors. These linguistic ratings can be expressed in trapezoidal or triangular fuzzy numbers. Then, a hierarchical MCDM method based on fuzzy-sets theory is proposed to deal with the fuel buses selection problem. To simultaneously determine the ranking order of all alternatives based on the concept of the TOPSIS by calculating the distances to the both fuzzy positive-ideal solution (FPIS) and fuzzy negative-ideal solution (FNIS), a closeness coefficient is defined. In the second presented method we extend preference selection index (PSI) method for fuzzy environment. In this method performance ratings of criteria evaluate by linguistic variable which can be expressed in trapezoidal fuzzy numbers. Finally, an example is shown to highlight the procedure of the proposed methods and compare the results of these methods with each other. [ABSTRACT FROM AUTHOR]

Abstract: Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. There has been some progress concerning practical applications of soft set theory, especially the use of soft sets in decision making. In this paper we generalize the adjustable approach to fuzzy soft sets based decision making. Concretely, we present an adjustable approach to intuitionistic fuzzy soft sets based decision making by using level soft sets of intuitionistic fuzzy soft sets and give some illustrative examples. The properties of level soft sets are presented and discussed. Moreover, we also introduce the weighted intuitionistic fuzzy soft sets and investigate its application to decision making. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we will present a motion pattern recognition based Kalman filter (PRKF), and apply it to the time difference of arrival (TDOA) algorithm of indoor localization. The state matrix in Kalman filter (KF) is determined by the motion pattern which the target node is supposed to act, and this will bring new system error if the assumption is not correct. Considering this, we first create three fuzzy sets using three KFs whose state matrix stand for different motion patterns, then linearly combined the memberships of a target node of the fuzzy sets. Finally, simulation results show that the PRKF can enhance the localization accuracy about more than 20%. [Copyright &y& Elsevier]

Abstract: Project scheduling problem is to determine the schedule of allocating resources so as to balance the total cost and the completion time. This paper considers a type of project scheduling problem with fuzzy activity duration times. According to some management goals, three types of fuzzy models are built to solve the project scheduling problem. Moreover, the technique of fuzzy simulation and genetic algorithm are integrated to design a hybrid intelligent algorithm to solve the fuzzy models. Finally, some numerical examples are given to illustrate the effectiveness of the algorithm. [Copyright &y& Elsevier]

Abstract: In this paper, the equilibrium optimization problem is proposed and the assignment problem is extended to the equilibrium multi-job assignment problem, equilibrium multi-job quadratic assignment problem and the minimum cost and equilibrium multi-job assignment problem. Furthermore, the mathematical models of the equilibrium multi-job assignment problem and the equilibrium multi-job quadratic assignment problem with fuzzy parameters are formulated. Finally, a genetic algorithm is designed for solving the proposed programming models and some numerical examples are given to verify the efficiency of the designed algorithm. [Copyright &y& Elsevier]

Abstract: This paper proposes a parametric programming approach to analyze the fuzzy maximum total return in the continuous knapsack problem with fuzzy objective weights, in that the membership function of the maximum total return is constructed. The idea is based on Zadeh’s extension principle, α-cut representation, and the duality theorem of linear programming. A pair of linear programs parameterized by possibility level α is formulated to calculate the lower and upper bounds of the fuzzy maximum total return at α, through which the membership function of the maximum total return is constructed. To demonstrate the validity of the proposed procedure, an example studied by the previous studies is investigated successfully. Since the fuzzy maximum total return is completely expressed by a membership function rather than by a crisp value reported in previous studies, the fuzziness of object weights is conserved completely, and more information is provided for making decisions in real-world resource allocation applications. The generalization of the proposed approach for other types of knapsack problems is also straightforward. [Copyright &y& Elsevier]

In this short paper, we point out the object-parameter method proposed in [1] is not always effective under some incomplete circumstances, sometimes, the predicted data obtained by this method are greater than one, this result does not satisfy what is required in the definition of membership degree of fuzzy sets. [ABSTRACT FROM AUTHOR]

Abstract: This paper proposes a procedure to construct the membership functions of the system characteristics of a redundant repairable system with two primary units and one standby in which the coverage factor is the same for an operating unit failure as that for a standby unit failure. Times to failure and times to repair of the operating and standby units are assumed to follow fuzzified exponential distributions. The α-cut approach is used to extract from the fuzzy repairable system a family of conventional crisp intervals for the desired system characteristics, determined with a set of parametric nonlinear programs using their membership functions. A numerical example is solved successfully to illustrate the practicality of the proposed approach. Because the system characteristics are governed by the membership functions, more information is provided for use by management, and because the redundant system is extended to the fuzzy environment, general repairable systems are represented more accurately and the analytic results are more useful for designers and practitioners. [Copyright &y& Elsevier]

Abstract: This paper develops a single wholesaler and multi retailers mixture inventory distribution model for a single item involving controllable lead-time with backorder and lost sales. The retailers purchase their items from the wholesaler in lots at some intervals throughout the year to meet the customers’ demand. Not to loose the demands, the retailers offer a price discount to the customers on the stock-out items. Here, it is assumed that the lead-time demands of retailers are uncertain in both stochastic and fuzzy sense, i.e., these are simultaneously random and imprecise. To implement this behavior of the lead-time demands, at first, these demands are assumed to be random, say following a normal distribution. With these random demands, the expected total cost for each retailer is obtained. Now, the mean lead-time demands (which are crisp ones) of the retailers are fuzzified. This fuzzy nature of the lead-time demands implies that the annual average demands of the retailers must be fuzzy numbers, suppose these are triangular fuzzy numbers. Using signed distance technique for defuzzification, the estimate of total costs for each retailer is derived. Therefore, the problem is reduced to optimize the crisp annual costs of wholesaler and retailers separately. The multi-objective model is solved using Global Criteria method. Numerical illustrations have been made with the help of an example taking two retailers into consideration. Mathematical analyses have been made for global pareto-optimal solutions of the multi-objective optimization problem. Sensitivity analyses have been made on backorder ratio and pareto-optimal solutions for wholesaler and different retailers are compared graphically. [Copyright &y& Elsevier]

Abstract: In this paper, we proposed a defuzzification using minimizer of the distance between the two fuzzy numbers. Then, we obtain the nearest point with respect to a fuzzy numbers and by considering the nearest point, we can present a ranking method for the fuzzy numbers. Also we give two new properties for ordering. Theorems and remarks are proposed for existence and uniqueness of the nearest point. The method is illustrated by numerical examples and compared with other methods. [Copyright &y& Elsevier]

Abstract: This paper develops a nonlinear programming approach to derive the membership functions of the steady-state performance measures in bulk arrival queueing systems with varying batch sizes, in that the arrival rate and service rate are fuzzy numbers. The basic idea is based on Zadeh’s extension principle. Two pairs of mixed integer nonlinear programs (MINLP) with binary variables are formulated to calculate the upper and lower bounds of the system performance measure at possibility level α. From different values of α, the membership function of the system performance measure is constructed. For practice use, the defuzzification of performance measures is also provided via Yager ranking index. To demonstrate the validity of the proposed method, a numerical example is solved successfully. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to correct two mistakes in [Appl. Math. Model. 35 (4) (2011) 1798–1809], which are: one of the properties of fuzzy rough set between two different universes and the definition of the upper approximation with the property for degree fuzzy rough set between two different universes. For the modified properties and the definition of upper approximation for degree fuzzy rough set between two different universes, we claim that the results obtained are correct and the errors have not further implications. [Copyright &y& Elsevier]