A new fuzzy additive model for determining the common set of weights in Data Envelopment Analysis

The main advantage of Data Envelopment Analysis (DEA) is that it does not require any priori weights for inputs and outputs and allows individual DMUs to evaluate their efficiencies with the input and output weights that are only most favorable weights for calculating their efficiency. It can be argued that if DMUs are experiencing similar circumstances, then the pricing of inputs and outputs should apply uniformly across all DMUs. That is using of different weights for DMUs makes their efficiencies unable to be compared and not possible to rank them on the same basis. This is a significant drawback of DEA; however literature observed many solutions including the use of common set of weights (CSW). Besides, the conventional DEA methods require accurate measurement of both the inputs and outputs; however, crisp input and output data may not relevant be available in real world applications. This paper develops a new model for the calculation of CSW in fuzzy environments using fuzzy DEA. Further, a numerical example is used to show the validity and efficacy of the proposed model and to compare the results with previous models available in the literature.

Fuzzy data envelopment analysis:a discrete approach

Data envelopment analysis (DEA) as introduced by Charnes, Cooper, and Rhodes (1978) is a linear programming technique that has widely been used to evaluate the relative efficiency of a set of homogenous decision making units (DMUs). In many real applications, the input-output variables cannot be precisely measured. This is particularly important in assessing efficiency of DMUs using DEA, since the efficiency score of inefficient DMUs are very sensitive to possible data errors. Hence, several approaches have been proposed to deal with imprecise data. Perhaps the most popular fuzzy DEA model is based on a-cut. One drawback of the a-cut approach is that it cannot include all information about uncertainty. This paper aims to introduce an alternative linear programming model that can include some uncertainty information from the intervals within the a-cut approach. We introduce the concept of "local a-level" to develop a multi-objective linear programming to measure the efficiency of DMUs under uncertainty. An example is given to illustrate the use of this method.

A fuzzy expected value approach under generalized data envelopment analysis

Fuzzy data envelopment analysis (DEA) models emerge as another class of DEA models to account for imprecise inputs and outputs for decision making units (DMUs). Although several approaches for solving fuzzy DEA models have been developed, there are some drawbacks, ranging from the inability to provide satisfactory discrimination power to simplistic numerical examples that handles only triangular fuzzy numbers or symmetrical fuzzy numbers. To address these drawbacks, this paper proposes using the concept of expected value in generalized DEA (GDEA) model. This allows the unification of three models - fuzzy expected CCR, fuzzy expected BCC, and fuzzy expected FDH models - and the ability of these models to handle both symmetrical and asymmetrical fuzzy numbers. We also explored the role of fuzzy GDEA model as a ranking method and compared it to existing super-efficiency evaluation models. Our proposed model is always feasible, while infeasibility problems remain in certain cases under existing super-efficiency models. In order to illustrate the performance of the proposed method, it is first tested using two established numerical examples and compared with the results obtained from alternative methods. A third example on energy dependency among 23 European Union (EU) member countries is further used to validate and describe the efficacy of our approach under asymmetric fuzzy numbers.

A stepwise fuzzy linear programming model with possibility and necessity relations

Linear programming (LP) is the most widely used optimization technique for solving real-life problems because of its simplicity and efficiency. Although conventional LP models require precise data, managers and decision makers dealing with real-world optimization problems often do not have access to exact values. Fuzzy sets have been used in the fuzzy LP (FLP) problems to deal with the imprecise data in the decision variables, objective function and/or the constraints. The imprecisions in the FLP problems could be related to (1) the decision variables; (2) the coefficients of the decision variables in the objective function; (3) the coefficients of the decision variables in the constraints; (4) the right-hand-side of the constraints; or (5) all of these parameters. In this paper, we develop a new stepwise FLP model where fuzzy numbers are considered for the coefficients of the decision variables in the objective function, the coefficients of the decision variables in the constraints and the right-hand-side of the constraints. In the first step, we use the possibility and necessity relations for fuzzy constraints without considering the fuzzy objective function. In the subsequent step, we extend our method to the fuzzy objective function. We use two numerical examples from the FLP literature for comparison purposes and to demonstrate the applicability of the proposed method and the computational efficiency of the procedures and algorithms. © 2013-IOS Press and the authors. All rights reserved.

A fuzzy AHP based integer linear programming model for the multi-criteria transshipment problem

Purpose – The purpose of this research is to develop a holistic approach to maximize the customer service level while minimizing the logistics cost by using an integrated multiple criteria decision making (MCDM) method for the contemporary transshipment problem. Unlike the prevalent optimization techniques, this paper proposes an integrated approach which considers both quantitative and qualitative factors in order to maximize the benefits of service deliverers and customers under uncertain environments. Design/methodology/approach – This paper proposes a fuzzy-based integer linear programming model, based on the existing literature and validated with an example case. The model integrates the developed fuzzy modification of the analytic hierarchy process (FAHP), and solves the multi-criteria transshipment problem. Findings – This paper provides several novel insights about how to transform a company from a cost-based model to a service-dominated model by using an integrated MCDM method. It suggests that the contemporary customer-driven supply chain remains and increases its competitiveness from two aspects: optimizing the cost and providing the best service simultaneously. Research limitations/implications – This research used one illustrative industry case to exemplify the developed method. Considering the generalization of the research findings and the complexity of the transshipment service network, more cases across multiple industries are necessary to further enhance the validity of the research output. Practical implications – The paper includes implications for the evaluation and selection of transshipment service suppliers, the construction of optimal transshipment network as well as managing the network. Originality/value – The major advantages of this generic approach are that both quantitative and qualitative factors under fuzzy environment are considered simultaneously and also the viewpoints of service deliverers and customers are focused. Therefore, it is believed that it is useful and applicable for the transshipment service network design.

A fuzzy decision tree-based duration model for Standard Yorùbá text-to-speech synthesis

In this paper, we present syllable-based duration modelling in the context of a prosody model for Standard Yorùbá (SY) text-to-speech (TTS) synthesis applications. Our prosody model is conceptualised around a modular holistic framework. This framework is implemented using the Relational Tree (R-Tree) techniques. An important feature of our R-Tree framework is its flexibility in that it facilitates the independent implementation of the different dimensions of prosody, i.e. duration, intonation, and intensity, using different techniques and their subsequent integration. We applied the Fuzzy Decision Tree (FDT) technique to model the duration dimension. In order to evaluate the effectiveness of FDT in duration modelling, we have also developed a Classification And Regression Tree (CART) based duration model using the same speech data. Each of these models was integrated into our R-Tree based prosody model. We performed both quantitative (i.e. Root Mean Square Error (RMSE) and Correlation (Corr)) and qualitative (i.e. intelligibility and naturalness) evaluations on the two duration models. The results show that CART models the training data more accurately than FDT. The FDT model, however, shows a better ability to extrapolate from the training data since it achieved a better accuracy for the test data set. Our qualitative evaluation results show that our FDT model produces synthesised speech that is perceived to be more natural than our CART model. In addition, we also observed that the expressiveness of FDT is much better than that of CART. That is because the representation in FDT is not restricted to a set of piece-wise or discrete constant approximation. We, therefore, conclude that the FDT approach is a practical approach for duration modelling in SY TTS applications. © 2006 Elsevier Ltd. All rights reserved.

A computational model of prosody for Yorùbá text-to-speech synthesis

This work examines prosody modelling for the Standard Yorùbá (SY) language in the context of computer text-to-speech synthesis applications. The thesis of this research is that it is possible to develop a practical prosody model by using appropriate computational tools and techniques which combines acoustic data with an encoding of the phonological and phonetic knowledge provided by experts. Our prosody model is conceptualised around a modular holistic framework. The framework is implemented using the Relational Tree (R-Tree) techniques (Ehrich and Foith, 1976). R-Tree is a sophisticated data structure that provides a multi-dimensional description of a waveform. A Skeletal Tree (S-Tree) is first generated using algorithms based on the tone phonological rules of SY. Subsequent steps update the S-Tree by computing the numerical values of the prosody dimensions. To implement the intonation dimension, fuzzy control rules where developed based on data from native speakers of Yorùbá. The Classification And Regression Tree (CART) and the Fuzzy Decision Tree (FDT) techniques were tested in modelling the duration dimension. The FDT was selected based on its better performance. An important feature of our R-Tree framework is its flexibility in that it facilitates the independent implementation of the different dimensions of prosody, i.e. duration and intonation, using different techniques and their subsequent integration. Our approach provides us with a flexible and extendible model that can also be used to implement, study and explain the theory behind aspects of the phenomena observed in speech prosody.

Ranking efficient decision-making units in data envelopment analysis using fuzzy concept

This paper introduces a new mathematical method for improving the discrimination power of data envelopment analysis and to completely rank the efficient decision-making units (DMUs). Fuzzy concept is utilised. For this purpose, first all DMUs are evaluated with the CCR model. Thereafter, the resulted weights for each output are considered as fuzzy sets and are then converted to fuzzy numbers. The introduced model is a multi-objective linear model, endpoints of which are the highest and lowest of the weighted values. An added advantage of the model is its ability to handle the infeasibility situation sometimes faced by previously introduced models.

Development of a probabilistic graphical structure from a model of mental health clinical expertise

This thesis explores the process of developing a principled approach for translating a model of mental-health risk expertise into a probabilistic graphical structure. Probabilistic graphical structures can be a combination of graph and probability theory that provide numerous advantages when it comes to the representation of domains involving uncertainty, domains such as the mental health domain. In this thesis the advantages that probabilistic graphical structures offer in representing such domains is built on. The Galatean Risk Screening Tool (GRiST) is a psychological model for mental health risk assessment based on fuzzy sets. In this thesis the knowledge encapsulated in the psychological model was used to develop the structure of the probability graph by exploiting the semantics of the clinical expertise. This thesis describes how a chain graph can be developed from the psychological model to provide a probabilistic evaluation of risk that complements the one generated by GRiST’s clinical expertise by the decomposing of the GRiST knowledge structure in component parts, which were in turned mapped into equivalent probabilistic graphical structures such as Bayesian Belief Nets and Markov Random Fields to produce a composite chain graph that provides a probabilistic classification of risk expertise to complement the expert clinical judgements

Efficiency measurement in fuzzy additive data envelopment analysis

Performance evaluation in conventional data envelopment analysis (DEA) requires crisp numerical values. However, the observed values of the input and output data in real-world problems are often imprecise or vague. These imprecise and vague data can be represented by linguistic terms characterised by fuzzy numbers in DEA to reflect the decision-makers' intuition and subjective judgements. This paper extends the conventional DEA models to a fuzzy framework by proposing a new fuzzy additive DEA model for evaluating the efficiency of a set of decision-making units (DMUs) with fuzzy inputs and outputs. The contribution of this paper is threefold: (1) we consider ambiguous, uncertain and imprecise input and output data in DEA, (2) we propose a new fuzzy additive DEA model derived from the a-level approach and (3) we demonstrate the practical aspects of our model with two numerical examples and show its comparability with five different fuzzy DEA methods in the literature. Copyright © 2011 Inderscience Enterprises Ltd.

An overall profit Malmquist productivity index with fuzzy and interval data

Although crisp data are fundamentally indispensable for determining the profit Malmquist productivity index (MPI), the observed values in real-world problems are often imprecise or vague. These imprecise or vague data can be suitably characterized with fuzzy and interval methods. In this paper, we reformulate the conventional profit MPI problem as an imprecise data envelopment analysis (DEA) problem, and propose two novel methods for measuring the overall profit MPI when the inputs, outputs, and price vectors are fuzzy or vary in intervals. We develop a fuzzy version of the conventional MPI model by using a ranking method, and solve the model with a commercial off-the-shelf DEA software package. In addition, we define an interval for the overall profit MPI of each decision-making unit (DMU) and divide the DMUs into six groups according to the intervals obtained for their overall profit efficiency and MPIs. We also present two numerical examples to demonstrate the applicability of the two proposed models and exhibit the efficacy of the procedures and algorithms. © 2011 Elsevier Ltd.

Aggregating preference ranking with fuzzy Data Envelopment Analysis

Selecting the best alternative in a group decision making is a subject of many recent studies. The most popular method proposed for ranking the alternatives is based on the distance of each alternative to the ideal alternative. The ideal alternative may never exist; hence the ranking results are biased to the ideal point. The main aim in this study is to calculate a fuzzy ideal point that is more realistic to the crisp ideal point. On the other hand, recently Data Envelopment Analysis (DEA) is used to find the optimum weights for ranking the alternatives. This paper proposes a four stage approach based on DEA in the Fuzzy environment to aggregate preference rankings. An application of preferential voting system shows how the new model can be applied to rank a set of alternatives. Other two examples indicate the priority of the proposed method compared to the some other suggested methods.

Fuzzy assessment of performance of a decision making units using DEA:a non-radical approach

Data Envelopment Analysis (DEA) is recognized as a modern approach to the assessment of performance of a set of homogeneous Decision Making Units (DMUs) that use similar sources to produce similar outputs. While DEA commonly is used with precise data, recently several approaches are introduced for evaluating DMUs with uncertain data. In the existing approaches many information on uncertainties are lost. For example in the defuzzification, the a-level and fuzzy ranking approaches are not considered. In the tolerance approach the inequality or equality signs are fuzzified but the fuzzy coefficients (inputs and outputs) are not treated directly. The purpose of this paper is to develop a new model to evaluate DMUs under uncertainty using Fuzzy DEA and to include a-level to the model under fuzzy environment. An example is given to illustrate this method in details.

Developing a probabilistic graphical structure from a model of mental-health clinical risk expertise

This paper explores the process of developing a principled approach for translating a model of mental-health risk expertise into a probabilistic graphical structure. The Galatean Risk Screening Tool [1] is a psychological model for mental health risk assessment based on fuzzy sets. This paper details how the knowledge encapsulated in the psychological model was used to develop the structure of the probability graph by exploiting the semantics of the clinical expertise. These semantics are formalised by a detailed specification for an XML structure used to represent the expertise. The component parts were then mapped to equivalent probabilistic graphical structures such as Bayesian Belief Nets and Markov Random Fields to produce a composite chain graph that provides a probabilistic classification of risk expertise to complement the expert clinical judgements. © Springer-Verlag 2010.

Intonation contour realisation for Standard Yorùbá text-to-speech synthesis:a fuzzy computational approach

This paper presents a novel intonation modelling approach and demonstrates its applicability using the Standard Yorùbá language. Our approach is motivated by the theory that abstract and realised forms of intonation and other dimensions of prosody should be modelled within a modular and unified framework. In our model, this framework is implemented using the Relational Tree (R-Tree) technique. The R-Tree is a sophisticated data structure for representing a multi-dimensional waveform in the form of a tree. Our R-Tree for an utterance is generated in two steps. First, the abstract structure of the waveform, called the Skeletal Tree (S-Tree), is generated using tone phonological rules for the target language. Second, the numerical values of the perceptually significant peaks and valleys on the S-Tree are computed using a fuzzy logic based model. The resulting points are then joined by applying interpolation techniques. The actual intonation contour is synthesised by Pitch Synchronous Overlap Technique (PSOLA) using the Praat software. We performed both quantitative and qualitative evaluations of our model. The preliminary results suggest that, although the model does not predict the numerical speech data as accurately as contemporary data-driven approaches, it produces synthetic speech with comparable intelligibility and naturalness. Furthermore, our model is easy to implement, interpret and adapt to other tone languages.