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To achieve carbon neutrality, the power structure is bound to usher in fundamental transformations. In this study, an energy–carbon–water nexus (F-ECWN) system model is developed to assess trade-offs among different decision-making objectives within a power system under varying policy orientations. The model incorporates forecast data derived from various global climate models, leverages neural network prediction, draws upon multi-criteria decision-making theory, and incorporates fuzzy theory. Moreover, to effectively handle a multitude of fuzzy parameters within the model, a novel dual-interval algorithm for solving fuzzy linear programming is proposed. The F-ECWN model has the capability to derive the fuzzy membership function for each variable within the model ensuring that errors remain below 1 %, reflecting the uncertainty stemming from diverse policy orientations and technology choices. Various scenarios were formulated to gauge the impact of policy orientation on the allocation decisions of regional energy systems. Additionally, sensitivity analysis has been conducted to assess the effects of uncertain parameters on modeling outputs. The results of the applied research in Fujian Province have revealed that the presence of uncertainties in the energy system's parameters can significantly influence model outputs and decision-making processes. Furthermore, the modeling results demonstrate the region's substantial potential for reducing carbon emissions. Under optimal policy guidance and climate conditions, the total carbon emissions can be reduced by 65 %, with a 36.14 % increase in the total system cost. These findings are anticipated to provide valuable support for formulating optimal decisions regarding regional energy-carbon-water nexus system and related environmental policies. [Display omitted] • A new algorithm for solving fuzzy multi-objective programming problems is proposed. • A robust fuzzy number reduction algorithm based on probabilistic integration is proposed. • E -C-W nexus in Fujian Province under different policy orientations is quantified. • Incorporation of CMIP6 forecast data into renewable energy parameters construction. • Carbon and water synergy effects in regional power system optimization are observed. [ABSTRACT FROM AUTHOR]

Many companies established a closed-loop supply chain to increase their supply chain profit and sustainability. The goal of this study is to design a faucet closed-loop supply chain network, in which forward supply chain is explained is aimed to increase the profit and environmental savings, and used faucet exchange plan is presented in the reverse chain. A mixed integer linear programming model under demand uncertainty is developed. A single-objective, multi-product and multi-period model is proposed for this network. The objective function maximizes the total profit along with optimization of the flow within facilities. Moreover, a fuzzy linear programming and fuzzy stochastic linear programming model describes the proposed network's decisions. Finally, to validate the proposed model, a real case study is performed on Tehrandoosh Company and the model solutions are analyzed. The obtained results show the performance of the presented network. In which, considering the exchange plan would lead to 58 percent increase in the total profit of the supply chain. Furthermore, this study provided efficient and reliable managerial solutions for the faucet industry decision makers. [ABSTRACT FROM AUTHOR]

Abstract Traditional games theory is based on the idea of having a set of deterministic payoffs to represent actions taken by homogeneous individuals (interaction among players) while in several real scenarios, social and business conflicts are handled by groups of actors with different technical/social skills. This way, a group-game problem can be seen as to two-groups of players who individually select a strategy to win, so the information given by every player in their group affects the payoffs of such game. Sometimes, deterministic payoffs are not available, and even when they can be obtained from negotiation between players, they are not clearly defined. Indeed within groups, individual players could have different perceptions of the payoffs for which we propose a uncertain based matrix games model involving Interval-valued fuzzy payoffs. [ABSTRACT FROM AUTHOR]

Abstract Recycling aims at preventing rapid depletion of natural resources while transforming produced waste into value for economy. However, this process becomes a major challenge in automotive industry, which requires cooperative engagement of multiple players within a complex supply chain. In line with the essence of the topic, government agencies around the world issue directives drawing regulatory frameworks for designing recycling operations comprising various activities such as collection of end-of-life vehicles (ELVs), recovery of reusable components, shredding ELV's body, recycling valuable materials and disposal of the hazardous waste. In general, the amount of returned product in a reverse logistics network is highly uncertain, and the ELV market in Turkey is no exception to this. For that purpose, this study aims developing a fuzzy mixed integer location-allocation model for reverse logistic network of ELVs conforming to the existing directives in Turkey. Accordingly, this study uses a novel approach and assumes that ELV supply in the network is uncertain. The merit of the proposed mathematical model is proved on a real world scenario addressing the reverse logistics design problem for ELVs generated in metropolitan area of Istanbul. The network generated specifies that recycling process is not profitable under the existing circumstances with the given level of supplied ELV and the returned product records per capita in Istanbul are far beyond the EU averages. Consequently, sensitivity analyses question the reliability of the obtained results. [ABSTRACT FROM AUTHOR]

The operation of bidirectional V2G participating in ancillary services markets is particularly challenging due to various uncertainties. This work proposes an algorithm to optimize the uncertain operation of bidirectional V2G for an electric vehicle (EV) aggregator. The proposed algorithm maximizes the profits of the aggregator while providing additional system flexibility and low charging costs to the EV owners. The proposed algorithm considers electricity market and EV mobility uncertainties using fuzzy linear programming. These uncertainties include those of the regulation and responsive reserve prices, deployment signals, and the energy used for EV trips. Unlike other works in the literature, the proposed algorithm is capable of considering a large number of uncertain parameters without greatly impacting the problem's complexity or simulation time. Also, the algorithm does not result in conservative solutions. The simulation results show the superiority of the proposed fuzzy algorithm over its deterministic counterpart in terms of higher realized profits. Also, the proposed approach results in lower real-time penalties, which ensures its superiority to provide the EVs with the required trip energy at the required time. In addition, it results in lower battery degradation costs, which helps increase the life time of the EVs. [ABSTRACT FROM AUTHOR]

This paper proposes a new energy management scheduling model for optimizing operation costs of a non-isolated micro-grid. It's assumed that the micro-grid includes some renewable energy resources like wind turbines and photovoltaic panels, fuel cell, and energy storage systems, and is connected to the macro-grid through a distributed transmission line with limited capacity. Different possible uncertainties associated with different elements of the micro-grid like forecasted outputs of renewable resources, a maximum capacity of the fuel cell and batteries, a maximum capacity of distributed line and hourly demands are considered in the proposed model. Fuzzy sets concept is used to model these uncertainties in the micro-grid and a three stage optimization method is applied to find the optimal scheduling of the micro-grid under the uncertainties. The proposed method is implemented to a real case study. Impacts of variations in each element of the micro-grid and its associated uncertainty on the results are investigated through a sensitivity analysis. [ABSTRACT FROM AUTHOR]

In this paper, efforts are made to find the fuzzy optimal solutions for electric tomography imaging (ETI) process. Most of the conventional ETI methods depend on an optimization process with a deterministic objective function and constraint conditions. The effective application of these methods requires the availability of measured data. However, in practical ETI possess, the assessable data are uncertain, imprecise, incomplete, and characterized by fuzziness. In this paper, the fuzzy characteristics hidden in the ETI process are extracted, and a new model of fuzzy optimization is built up, whose optimal solution is obtained by using fuzzy linear programming (FLP). The FLP model can provide a robust methodology to boost the ETI spatial resolution. Thus it creates a new and promising research direction for the optimization of the ETI process. Experimental results prove that compared with other existing ETI algorithms, the proposed method is more effective in reconstructing complex and discretely distributed objects. [ABSTRACT FROM AUTHOR]

As a result of uncertainty and complexity for environments of decision-making, it is more suitable for decision makers to use hesitant fuzzy linguistic information. In this paper, a novel group decision making (GDM) model based on fuzzy linear programming is proposed for incomplete comparative expressions with hesitant fuzzy linguistic term set (HFLTSs). We establish an equivalence theorem of additive consistency between 2-tuple fuzzy linguistic preference relation (FLPR) and corresponding fuzzy preference relation. Based on this framework, a fuzzy linear programming is established to address incomplete comparative expressions with HFLTSs. It is more important that the proposed fuzzy linear programming has a double action, finding the highest consistent incomplete 2-tuple FLPR and increasing inconsistent 2-tuple FLPR to the additive consistent 2-tuple FLPR based on given incomplete comparative expressions with HFLTSs. By this means, a novel GDM model is constructed based on importance induced ordered weighted averaging operator. Finally, an investment decision-making in real-world is solved by the proposed model, which shows the result of GDM is effectiveness. [ABSTRACT FROM AUTHOR]

Most of the existing methods for solving fully fuzzy mathematical programs are based on the standard fuzzy arithmetic operations and/or Zadeh's extension principle. These methods may produce questionable results for many real-life applications. Due to this fact, this paper presents a novel method based on the constrained fuzzy arithmetic concept to solve fully fuzzy balanced/unbalanced transportation problems in which all of the parameters (source capacities, demands of destinations, transportation costs etc.) as well as the decision variables (transportation quantities) are considered as fuzzy numbers. In the proposed method, the requisite crisp and/or fuzzy constraints between the base variables of the fuzzy components are provided from the decision maker according to his/her exact or vague judgments. Thereafter, fuzzy arithmetic operations are performed under these requisite constraints by taking into account the additional information while transforming the fuzzy transportation model into crisp equivalent form. Therefore, various fuzzy efficient solutions can be generated by making use of the proposed method according to the decision maker's risk attitude. In order to present the efficiency/applicability of the proposed method, different types of fully fuzzy transportation problems are generated and solved as illustrative examples. A detailed comparative study is also performed with other methods available in the literature. The computational analysis have shown that relatively more precise solutions are obtained from the proposed method for “risk-averse” and “partially risk-averse” decision makers. The proposed method also successfully provided fuzzy acceptable solutions for “risk seekers” with high degree of uncertainty similar to the other existing methods in the literature. [ABSTRACT FROM AUTHOR]

As the retailing industry becomes more customer oriented, it struggles with integrating the voice-of-customers into quality development policies, determining accurate customer expectations, and understanding how to incorporate the required store attributes in retailing activities. The aim of this study is to provide managers with a more decisive and sustainable framework to fulfill customer satisfaction by determining the most essential customer needs and gain a competitive advantage by applying a benchmarking process for the whole retailing activities. To essentially support managers in determining and implementing required store attributes, this study develops a sustainable linear programming (LP) based Quality Function Deployment (QFD) methodology under IVIF-environment. The proposed method determines more accurate customer expectations (CEs) and related service requirements (SRs). Accordingly, while "Clothing Quality", Price Policy", and "Staff Behavior" are determined as the most important CEs, "Design of Customer Persona", "Production Cost", and "Marketing Applications" are obtained as the most affecting SRs. Since no specific study in the literature addresses uncertainty in CEs and SRs in the apparel retailing industry, we developed an LP-based QFD under the IVIF-environment framework, which reflects the ambiguity and vagueness of the evaluations better. Thus, this study contributes to the literature by proposing a sustainable framework for managers to make decisions that are more effective and take sustainable actions. The companies who want to get the advantage in the apparel retailing industry should follow the methodology provided within this study by adding their business specific dimensions. Lastly, to represent the validity and feasibility of the proposed approach sensitivity and comparison analysis are conducted. The results of comparison show that the LP based QFD method is as consistent as other method but more effective in terms of handling ambiguity and fuzziness of expert evaluations, comprehensively. [ABSTRACT FROM AUTHOR]

For power system decision makers, they should decide a reasonable wind capacity or target penetration level considering the balance of wind curtailment, investment, system reliability and social benefit. Assessing wind curtailment under different wind capacity, considering the wind resource uncertainty, is therefore necessary for utilizing wind power effectively. In this paper, fuzzy wind capacity factor is proposed to analyze and propagate the uncertainty of wind resources based on the wind speed distribution. By investigating the maximal wind power output that can be accepted in candidate system nodes, a fuzzy linear programming problem is proposed to decide the required wind capacity considering the fuzzy wind capacity factor. Based on the cumulative distribution function of wind capacity resulting from large number of simulations, the relationship between wind capacity and wind curtailment is determined. The effectiveness of the proposed model is validated by its application in two power grids. The results demonstrate that the proposed model and method are capable of describing the possibilistic uncertainty of wind resources and assessing wind curtailment under different wind capacity. Compared with previous research, this method can aid power system planners to allocate wind capacity considering the possibilistic uncertainty of wind resources. [ABSTRACT FROM AUTHOR]

Fuzzy Linear Programming is among the best and most studied topics in the Fuzzy Sets and Systems area. In this paper, the author describes the main developments, results and solution methods achieved in this topic throughout the last 35 years, from a personal point of view. Short descriptions on different Fuzzy Linear Programming problems are provided and future research lines are briefly pointed out. [ABSTRACT FROM AUTHOR]

The first researchers to publish an operationalization of the Bellman and Zadeh's original approach to “Decision making in a fuzzy environment” [1] were H. Tanaka and K. Asai in their 1973 article [2] which was published in English in 1974 [3] . These two researchers developed many fundamental theories and methods including novel concepts of fuzzy operations research [4] with their collaborators. They went beyond the original proposal for fuzzy optimization set forth by Bellman/Zadeh. They are important founders of fuzzy optimization. This paper gives an overview of their great contributions to fuzzy optimization. [ABSTRACT FROM AUTHOR]

Linear programming (LP) is a widely used optimization method for solving real-life problems because of its efficiency. Although precise data are fundamentally indispensable in conventional LP problems, the observed values of the data in real-life problems are often imprecise. Fuzzy sets theory has been extensively used to represent imprecise data in LP by formalizing the inaccuracies inherent in human decision-making. The fuzzy LP (FLP) models in the literature generally either incorporate the imprecisions related to the coefficients of the objective function, the values of the right-hand-side, and/or the elements of the coefficient matrix. We propose a new method for solving FLP problems in which the coefficients of the objective function and the values of the right-hand-side are represented by symmetric trapezoidal fuzzy numbers while the elements of the coefficient matrix are represented by real numbers. We convert the FLP problem into an equivalent crisp LP problem and solve the crisp problem with the standard primal simplex method. We show that the method proposed in this study is simpler and computationally more efficient than two competing FLP methods commonly used in the literature. [ABSTRACT FROM AUTHOR]

Considering various situations and characteristics of supply chain management, we regard the outsourcing provider selection as a type of fuzzy inhomogenous multiattribute group decision making (MAGDM) problems with fuzzy alternatives' comparisons and incomplete weight information. Hereby we focus on developing a new fuzzy linear programming method for solving such MAGDM problems. In this method, the decision makers' preferences are given through pair-wise alternatives' comparisons with fuzzy truth degrees represented as trapezoidal fuzzy numbers (TrFNs). Intuitionistic fuzzy sets, TrFNs, intervals and real numbers are used to express the inhomogenous decision information. Under the condition that the fuzzy positive ideal solution (PIS) and fuzzy negative ideal solution (NIS) are known, the fuzzy consistency and inconsistency indices are defined on the basis of the relative closeness degrees and expressed with TrFNs. The attribute weights are estimated through constructing a new fuzzy linear programming model, which is solved by the developed method of fuzzy linear programming with TrFNs. Through solving the constructed linear goal programming model, we obtain the collective comprehensive relative closeness degrees of alternatives to the fuzzy PIS, which are used to rank the alternatives. The effectiveness of the proposed method is verified with an example of IT outsourcing provider selection. [ABSTRACT FROM AUTHOR]

Highlights: [•] Construct new fuzzy multiobjective programming models with TrFNs. [•] Propose a new method for the derived fuzzy linear programming models with TrFNs. [•] A method is developed to heterogeneous MADM based on fuzzy alternative comparison. [•] Weights are obtained solving the derived fuzzy linear programming model with TrFNs. [ABSTRACT FROM AUTHOR]

Abstract: Fuzzy linear programming with trapezoidal fuzzy numbers (TrFNs) is considered and a new method is developed to solve it. In this method, TrFNs are used to capture imprecise or uncertain information for the imprecise objective coefficients and/or the imprecise technological coefficients and/or available resources. The auxiliary multi-objective programming is constructed to solve the corresponding possibility linear programming with TrFNs. The auxiliary multi-objective programming involves four objectives: minimizing the left spread, maximizing the right spread, maximizing the left endpoint of the mode and maximizing the middle point of the mode. Three approaches are proposed to solve the constructed auxiliary multi-objective programming, including optimistic approach, pessimistic approach and linear sum approach based on membership function. An investment example and a transportation problem are presented to demonstrate the implementation process of this method. The comparison analysis shows that the fuzzy linear programming with TrFNs developed in this paper generalizes the possibility linear programming with triangular fuzzy numbers. [Copyright &y& Elsevier]

Abstract: The aim of this article is to introduce a formulation of fuzzy linear programming problems involving the level -interval-valued trapezoidal fuzzy numbers as parameters. Indeed, such a formulation is the general form of trapezoidal fuzzy number linear programming problems. Then, it is demonstrated that study of the sensitivity analysis for the level -interval-valued trapezoidal fuzzy number linear programming problems gives rise to the same expected results as those obtained for trapezoidal fuzzy number linear programming problems. [Copyright &y& Elsevier]

Abstract: To the best of our knowledge, there is no method in literature for solving such fully fuzzy linear programming (FLP) problems in which some or all the parameters are represented by unrestricted L-R flat fuzzy numbers. Also, to propose such a method, there is need to find the product of unrestricted L-R flat fuzzy numbers. However, there is no method in the literature to find the product of unrestricted L-R flat fuzzy numbers. In this paper, firstly the product of unrestricted L-R flat fuzzy numbers is proposed and then with the help of proposed product, a new method (named as Mehar’s method) is proposed for solving fully FLP problems. It is also shown that the fully FLP problems which can be solved by the existing methods can also be solved by the Mehar’s method. However, such fully FLP problems in which some or all the parameters are represented by unrestricted L-R flat fuzzy numbers can be solved by Mehar’s method but can not be solved by any of the existing methods. [Copyright &y& Elsevier]

Abstract: We present an interactive multi-criteria procedure that uses user defined tradeoff-cutting planes to identify promising feasible solution search space. New solutions in the promising feasible solution search space are constructed using combination of scatter and random search. The procedure of identifying tradeoff-cutting planes and scatter search continues for either a predetermined fixed number of iterations or until no solutions in the promising feasible solution search space are found. We formulate a coal production planning problem with fuzzy profit and fuzzy coal quality decision-maker utilities, and apply our procedure for additive and multiplicative decision-maker utilities. [Copyright &y& Elsevier]

Abstract: This paper shows a general model of a Mixed Production Planning problem with fuzzy demands. The main focus is the development of a model for Production Planning using fuzzy sets in order to use classical mathematical programming techniques to reach an optimal solution over a multiple criteria context. The classical Fuzzy Linear Programming model namely the Soft Constraints Model is used to involve flexibility in the problem. Moreover, an Interval Fuzzy Set approach is used to involve uncertainty in the problem. [Copyright &y& Elsevier]

Abstract: Quadratic programming can be seen both as a general approach to linear programming and a special class of non-linear programming. Moreover, quadratic programming problems are of utmost importance in an increasing variety of practical fields, such as, regression, efficient production and portfolio selection. As ambiguity and vagueness are natural and ever-present in real-life situations requiring solutions, it makes perfect sense to attempt to address them using fuzzy quadratic programming problems. The main purpose of this paper is to study the strictly sensitivity analysis for fuzzy quadratic programming when simultaneously and independently variations occur in the right-hand-side of the constraints and the coefficients of the objective function. One presents computable auxiliary problems to identify the invariance intervals and give a fuzzy quadratic form of the optimal value function too. Some numerical examples are presented to illustrate the proposed method. [Copyright &y& Elsevier]

Abstract: The equivalence between the interval-valued fuzzy set (IVFS) and the intuitionistic fuzzy set (IFS) is exploited to study linear programming problems involving interval uncertainty modeled using IFS. The non-membership of IFS is constructed with three different viewpoints viz., optimistic, pessimistic, and mixed. These constructions along with their indeterminacy factors result in S-shaped membership functions in the fuzzy counterparts of the intuitionistic fuzzy linear programming models. The solution methodology of Yang et al. , and its subsequent generalization by Lin and Chen are used to compute the optimal solutions of the three fuzzy linear programming models. [Copyright &y& Elsevier]

Abstract: Selecting the best transportation investment project (TIP) is often a difficult task, since many social, environmental and economic criteria have to be considered simultaneously. Evaluating a set of different projects, especially the best set of alternatives, portfolios, is even more complex. Pursuing the goal of selecting the best TIP portfolio, we propose a fuzzy assessment method to aid the selection process of a multi-criterion project by utilizing the concept of entropy and interval normalization procedure in a fuzzy analytic hierarchy process (F-AHP). Then, regarding this informative phase, we propose a fuzzy linear programming model to select the best TIP portfolio under uncertain cost pressure. A real case study is conducted to illustrate the efficiency of the proposed method. [Copyright &y& Elsevier]

Abstract: We propose Fuzzy UTASTAR, a method for inferring fuzzy utility functions from a partial preorder of options evaluated on multiple criteria. It is an extension of the well-known UTASTAR method capable to handle both ordinary (crisp) and fuzzy evaluation data. This property gives much flexibility to decision makers because the majority of real-life decision problems involve a considerable level of uncertainty that hinders them from assigning exact evaluations (scores) to options. In case all evaluation data are crisp the method behaves exactly as the original UTASTAR. The proposed method builds fuzzy additive value functions taking as input a partial preorder on a subset of the options, called reference set, along with their associated scores on the criteria. The resulting fuzzy utility functions can subsequently be used to estimate the (fuzzy) utility of each option, thus allowing their ranking, prioritization, selection or classification. The ranking of the options in partial preorder is as compatible as possible to the original one. The method is implemented into a decision support system and is applied to an example from the transportation domain. Results are found to be in concordance with those of the original method. To the best of our knowledge this is the first attempt to extend UTASTAR method to handle both crisp and fuzzy evaluation data. [Copyright &y& Elsevier]

Abstract: In a recent paper, Ganesan and Veermani [K. Ganesan, P. Veeramani, Fuzzy linear programs with trapezoidal fuzzy numbers, Ann. Oper. Res. 143 (2006) 305–315] considered a kind of linear programming involving symmetric trapezoidal fuzzy numbers without converting them to the crisp linear programming problems and then proved fuzzy analogues of some important theorems of linear programming that lead to a new method for solving fuzzy linear programming (FLP) problems. In this paper, we obtain some another new results for FLP problems. In fact, we show that if an FLP problem has a fuzzy feasible solution, it also has a fuzzy basic feasible solution and if an FLP problem has an optimal fuzzy solution, it has an optimal fuzzy basic solution too. We also prove that in the absence of degeneracy, the method proposed by Ganesan and Veermani stops in a finite number of iterations. Then, we propose a revised kind of their method that is more efficient and robust in practice. Finally, we give a new method to obtain an initial fuzzy basic feasible solution for solving FLP problems. [Copyright &y& Elsevier]

Abstract: We present a new methodology for the analysis of fuzzy payoff matrix games. The main difficulty that appears in the study of these games is the comparison between the payoff values associated to the strategies of the players because these payoffs are fuzzy quantities. Our approach does not transform the fuzzy payoffs to crisp numbers via standard defuzzyfication but we use standard fuzzy orders which allows us to find solutions within the same space of fuzzy numbers. Moreover, we provide a method to solve these games finding equivalent fuzzy linear programs whose maximal solutions give the solutions of the games. [Copyright &y& Elsevier]

Abstract: Passenger coaches are one of the most important mass transportation vehicles for inner city passenger transportation and the proper interior design problem of the passenger coaches affect both passenger satisfaction and service quality. Defining user expectation and meeting them is important to solve this problem, which naturally contains ambiguity and vagueness. The study aims to maximize the total passenger number together with providing user satisfaction while designing interior area of passenger coaches. User expectations are defined by a survey and new design is defined with respect to anthropometric measures of human body. The collected data used to establish a fuzzy mathematic model to capture the ambiguity and vagueness of the problem and the mathematical model is solved using fuzzy optimization with respect to the constraints. [Copyright &y& Elsevier]

Abstract: The purpose of the paper is to propose a hybrid approach for asset allocation with simultaneous consideration of suitability and optimality. In this approach, recourse is taken to multiple methodologies to simultaneously capture the impersonal characteristics of the financial assets as well as the personal preferences of the investors. Accordingly, the paper draws on Investors’ survey to capture their preferences, cluster analysis to categorize the financial assets, the analytical hierarchy process for obtaining local weights (performance scores) of the financial assets corresponding to the four key asset allocation criteria and fuzzy multi-objective linear programming model for portfolio selection. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the best approximation for fuzzy valued functions using Chebyshev nodes. We prove a result on the best near-minimax approximation in the fuzzy sense. As an application, Runge’s phenomenon is fuzzified in two different cases, i.e. the best approximation and the best near-minimax approximation. [Copyright &y& Elsevier]

Abstract: The most decisive factor that survives enterprises under stiff competition is the development of new product (NPD), and when entering the product development stage after the fuzzy front end, a best project portfolio should be finalized in order to potentially create expected revenue and competitive advantage. However, even it reaches the end of the fuzzy front stage; the NPD project is still significantly involved with uncertainties, complexities and fuzziness. To assist R&D managers making decision in this environment, this study proposes a new approach which combines fuzzy set theory and multi-criteria group decision making method into a NPD project portfolio selection model. This model takes into account project performance, project delivery and project risk, and formulates the selection decision of NPD project portfolio as a fuzzy linear programming problem. The illustrative example shows that the model proposed can generate projects with the highest success rate under limited resources and manpower. [Copyright &y& Elsevier]

Abstract: The increasing popularity of employing web services for distributed systems contributes to the significance of service discovery. However, duplicated and similar functional features existing among services require service consumers to include additional aspects to evaluate the services. Generally, the service consumers would have different view on the quality of service (QoS) of service attributes. How to select the best composite service in theory among available service (WS) candidates for consumers is an interesting practical issue. This work proposes a QoS-aware service selection model based on fuzzy linear programming (FLP) technologies, in order to identify their dissimilarity on service alternatives, assist service consumers in selecting most suitable services with consideration of their expectations and preferences. This approach can obtain the optimal solution of consensual weight of QoS attribute and fuzzy positive ideal solution (FPIS) by extending LINMAP method, developed by Srinivasan and Shocker. Finally, two numerical examples are given to demonstrate the process of QoS-aware web service selection. The experimental results demonstrated that it is a feasible and supplementary manner in selecting the of web services. [Copyright &y& Elsevier]

Abstract: A fuzzy linear programming extension of the general life cycle model is described. The model is suitable for applications involving system design and optimization wherein fuzzy target levels for environmental flows or impacts are specified, and wherein multiple technological alternatives capable of providing equivalent products or services are present within the life cycle boundaries. Solving the model results in an optimal mix of technologies that embodies the best compromise of the fuzzy environmental targets specified. The model formulation is concise and consistent with the generalized matrix-based LCA model; its linearity also makes identification of the optimal solution straightforward. Applications of the model to two illustrative case studies are also given. [Copyright &y& Elsevier]

Abstract: This paper proposes an optimal design of single tuned shunt passive filter for the Underground Ramsis Metro Substation (URMS) using linear programming technique as a conventional technique and fuzzy linear programming (FLP) technique as a modern technique. The optimal filter parameters are obtained to overcome the harmonics generated in the voltage and current waveforms, due to the non-linear characteristics of the rectifiers, which distort the voltage and current waveforms at the point of common coupling (PCC). The harmonic levels are analyzed at different loading conditions using the fast Fourier transform (FFT) on MATLAB optimization program. The proposed optimal design of harmonic filter is based on minimizing the cost of the filter parameters and its losses. Both techniques are applied on the system under study. From the results, the THD levels of the line current and the bus voltage are reduced to be within the standard limits. A comparison between the optimal design filters is carried out using the LP and FLP techniques. However, the lowest cost of filter is obtained using the FLP technique. [Copyright &y& Elsevier]

Abstract: Cardholders’ behavior prediction is an important issue in credit card portfolio management. As a promising data mining approach, multiple criteria programming (MCLP) has been successfully applied to classify credit cardholders’ behavior into two groups. In order to better control credit risk for financial institutes, this paper proposes three methods based on MCLP to improve the “Bad” catching accuracy rate. One is called MCLP with unbalanced training set selection, the second is called fuzzy linear programming (FLP) method with moving boundary, and the third is called penalized multi criteria linear programming (PMCLP). The experimental examples demonstrate the promising performance of these methods. [Copyright &y& Elsevier]

Abstract: In the article [G.Q. Zhang, Y.H. Wu, M. Remias, Jie Lu, Formulation of fuzzy linear programming problems as four-objective constrained optimization problems, Applied Mathematics and Computation 139 (2003) 383–399], Guangquan Zhang et al. proposed a number of theorems so as to convert the fuzzy linear programming (FLP) into a multi-objective optimization problems with four-objective functions. However there is a error in the article. [Copyright &y& Elsevier]

Abstract: One of the interesting, important and attractive problems in applied mathematics is approximation of functions in a given space. In this paper the problem is considered for fuzzy data and fuzzy functions using the defuzzification function of Fortemps and Roubens. Approximation of a fuzzy function on some given points for i =1,2,…, m is considered by some researchers as interpolation problem. But in interpolation problem we find a polynomial of degree at most n = m −1 where m is the number of points. But when we have lots of points (m is very large) it is not good or even possible to find such polynomials. In this case we want to find a polynomial of arbitrary degree which is an approximation to original function. One of the works has done is regression by some researchers and we introduced a different method. In this case we have m points but we want to find a polynomial of degree at most n < m but not n = m −1 necessarily. We introduce a fuzzy polynomial approximation as universal approximation of a fuzzy function on a discrete set of points and we present a method to compute it. We show that this approximation can be non-unique, however we choose one of them with the smallest amount of fuzziness. [Copyright &y& Elsevier]

Abstract: A class of fuzzy linear programming (FLP) problems with fuzzy coefficients based on fuzzy relations is introduced, the concepts of feasible and -maximal and minimal solutions are defined. The class of crisp (classical) LP problems can be embedded into the class of FLP ones. Moreover, for FLP problems a new concept of duality is introduced and the weak and strong duality theorems are derived. The introduced concepts and results are illustrated and discussed on a simple numerical example. [Copyright &y& Elsevier]

Abstract: Life cycle assessment (LCA) is known to entail multiple objective decision-making in the analysis of tradeoffs between different environmental impacts. The work of Azapagic and Clift in the late 1990s illustrates the use of multiple objective linear programming (MOLP) in the context of LCA. However, it will be noted that their approach yields a range of Pareto optimal alternatives from which the decision-maker must ultimately select the final solution. An alternative approach making use of Zimmermann''s symmetric fuzzy linear programming method is proposed and illustrated with a simple case study. The procedure effectively yields a single, optimal compromise solution. [Copyright &y& Elsevier]

Abstract: In this paper, the notions of subgradient, subdifferential, and differential with respect to convex fuzzy mappings are investigated, which provides the basis for the fuzzy extremum problem theory. We consider the problems of minimizing or maximizing a convex fuzzy mapping over a convex set and develop the necessary and/or sufficient optimality conditions. Furthermore, the concept of saddle-points and minimax theorems under fuzzy environment is discussed. The results obtained are used to formulate the Lagrangian dual of fuzzy programming. Under certain fuzzy convexity assumptions, KKT conditions for fuzzy programming are derived, and the “perturbed” convex fuzzy programming is considered. Finally, these results are applied to fuzzy linear programming and fuzzy quadratic programming. [Copyright &y& Elsevier]

Abstract: Composite power system reliability evaluation refers to assessment that considers both, generation and transmission facilities. The conventional dc load flow-based crisp linear programming (CLP) model used in composite power system reliability evaluation is modified as a fuzzy linear programming (FLP) model including fuzzy constraints and objective functions. This fuzzy linear programming model can include uncertainties that exist in certain variables and overcome the limitations of minor constraint violations in crisp linear programming model. This fuzzy optimization model is employed to test the system outage contingencies and to determine the degree of difficulty due to these contingencies. Fuzzy-based adequacy indices are obtained after all the selected contingencies have been analyzed using contingency enumeration approach. The effectiveness of the developed model is tested on the IEEE-14 bus system. [Copyright &y& Elsevier]

In this paper, a fuzzy comparison of fuzzy numbers is defined and a slack-based measure (SBM model) in data envelopment analysis (DEA) is extended to be a fuzzy DEA model, using it. Proposed measure is employed for evaluation and ranking of all decision making units, using a fuzzy concept called fuzzy profit. Also, it is shown that the introduced model is convenient for using weights restrictions. Furthermore, we compare the results of proposed model with Guo and Tanaka''s results [Fuzzy Sets Syst. 119 (2001) 149] by representing a numerical example introduced by them. [Copyright &y& Elsevier]

This paper develops a model for determining locations of undesirable facilities. It is formulated as multi-objective since the problem of locating undesirable facilities faces many conflicting criteria. A method is also proposed to appropriately address uncertainty associated with this class of location problems. The methodology developed in this study is tested using the real-world data. [Copyright &y& Elsevier]

Decisions made by the experts in the construction industry are usually approximate and contain some sort of imprecision. Classical linear programming (LP) model optimize the decision making situation in a crisp environment. It is difficult to get an optimum decision with imprecise information of the project environment using LP. In the construction industry, identifying optimum number of construction pieces of equipment require experts'' knowledge. When certain degree of flexibility needs to be incorporated in the given model to get more realistic results, fuzzy LP is used. But when the parameters on constraints and objective function are in a state of ambiguity then the extension principle is best suited, which is based on personal opinions and subjective judgments. The objective of this paper is to identify the optimum number of pieces of equipment required to complete the project in the targeted period with fuzzy data. A realistic case study has been considered for optimization and LINGO6 has been used to solve the various non-linear equations. [Copyright &y& Elsevier]

Quality function deployment (QFD) is the product development process to maximize customer satisfaction. The engineering design characteristics related to product performance are specified for this purpose. For dealing with the fuzzy nature in the product design processes, fuzzy approaches are applied to represent the relationships between customer requirements (CRs) and engineering design requirements (DRs) as well as among the DRs. A new measure for evaluating the fuzzy normalized relationships is derived. A fuzzy model is formulated to determine the fulfillment level of each DR for maximizing the customer satisfaction under the resource limitation and the considerations of technical difficulty and market competition. The producing ranges of fulfillment level of each DR and those of customer satisfaction can provide the QFD team with more information. An example is used to illustrate the model. [Copyright &y& Elsevier]

In this paper, we introduce a class of fuzzy linear programming problems and define the concepts of feasible and satisficing solutions—the necessary tools for dealing with such problems. In this way, we show that the class of crisp (classical) LP problems can be embedded into the class of FLP ones. Moreover, for FLP problems we define the concept of duality and prove the weak and strong duality theorems. Further, we define a class of interval linear programming problems as a special subclass of FLP problems and apply the previous results to this special case. [Copyright &y& Elsevier]

This paper presents a parametric approach for duality in fuzzy multi-criteria and multi-constraint level linear programming (MC2LP) which extends fuzzy linear programming approaches. First, the MC2-simplex method is used to solve the crisp prima–dual MC2LP pair and then, through these crisp formulations, separate membership functions are constructed for fuzzy primal and dual program by considering the corresponding primal and dual decisions. For each program, a set of fuzzy potential solutions is determined in terms of the membership function and the related optimal solution with a certain range of decision parameters. Finally, using the primal and dual membership functions, a fuzzy weak duality function is obtained for any pair of primal and dual fuzzy potential solutions. [Copyright &y& Elsevier]

Comparing fuzzy numbers, using the possibility programming approach, was presented by Negi and Lee [1]. In this paper, we are going to state some comments concerning their approach. The case of comparing different fuzzy numbers is considered. Also, the formulation of the crisp problem in the case of fuzzy linear programming and fuzzy multiobjective linear programming is discussed, and a new modified model, in each case, is stated. Our proposed approach is illustrated by numerical examples. [Copyright &y& Elsevier]