A Comparative Study of Chi-Square Goodness-of-Fit Under Fuzzy Environments

Testing goodness-of-fit plays a vital role in data analysis.  This problem seems to be much more complicated in the presence of vague data.  In this paper, the chi-square goodness-of-fit under trapezoidal fuzzy numbers (tfns.) is proposed using alpha cut interval method.  And the ranking grades of tfns. are also used to compute the chi-square test statistic.  The proposed technique is illustrated with two different numerical examples along with different methods of ranking grades for a concrete comparative study. Keywords: Chi-square Test, Fuzzy Sets, Trapezoidal Fuzzy Numbers, Alpha Cut, Ranking Function, Graded Mean Integration Representation

A Comparative Study of Latin Square Design Under Fuzzy Environments Using Trapezoidal Fuzzy Numbers

This paper deals with the problem of Latin Square Design (LSD) test using Trapezoidal Fuzzy Numbers (Tfns.).  The proposed test is analysed under various types of trapezoidal fuzzy models such as Alpha Cut Interval, Membership Function, Ranking Function, Total Integral Value and Graded Mean Integration Representation.  Finally a comparative view of the conclusions obtained from various test is given.  Moreover, two numerical examples having different conclusions have been given for a concrete comparative study.   Keywords: LSD, Trapezoidal Fuzzy Numbers, Alpha Cut, Membership Function, Ranking Function, Total Integral Value, Graded Mean Integration Representation.   AMS Mathematics Subject Classification (2010): 62A86, 62F03, 97K8

A single period inventory model for incorporating two-ordering opportunities under imprecise demand information

The ordering strategy for a single period inventory model is the key to achieve success in the competitive business environment. This article considers demand in a form of fuzzy number and discusses the SPIM in which the retailer has the opportunity to reorder once during the period. The entire period/season is divided into two slots and the reorder is to be made during the mid-season after the early-season demand has been observed. The objective is to find the expected optimal order quantity together with profit maximization. We illustrate the implementation of the proposed model using a numerical example and explain that the explicit consideration of this reordering opportunity could lead us to better results in terms of profitability

Evaluating high risks in large-scale projects using an extended VIKOR method under a fuzzy environment

The complexity of large-scale projects has led to numerous risks in their life cycle. This paper presents a new risk evaluation approach in order to rank the high risks in large-scale projects and improve the performance of these projects. It is based on the fuzzy set theory that is an effective tool to handle uncertainty. It is also based on an extended VIKOR method that is one of the well-known multiple criteria decision-making (MCDM) methods. The proposed decision-making approach integrates knowledge and experience acquired from professional experts, since they perform the risk identification and also the subjective judgments of the performance rating for high risks in terms of conflicting criteria, including probability, impact, quickness of reaction toward risk, event measure quantity and event capability criteria. The most notable difference of the proposed VIKOR method with its traditional version is just the use of fuzzy decision-matrix data to calculate the ranking index without the need to ask the experts. Finally, the proposed approach is illustrated with a real-case study in an Iranian power plant project, and the associated results are compared with two well-known decision-making methods under a fuzzy environment

One-Factor ANOVA Model Using Trapezoidal Fuzzy Numbers Through Alpha Cut Interval Method

Most of our traditional tools in descriptive and inferential statistics is based on crispness (preciseness) of data, measurements, random variable, hypotheses, and so on.  By crisp we mean dichotomous that is, yes-or-no type rather than more-or-less type.  But there are many situations in which the above assumptions are rather non-realistic such that we need some new tools to characterize and analyze the problem.  By introducing fuzzy set theory, different branches of mathematics are recently studied.  But probability and statistics attracted more attention in this regard because of their random nature.  Mathematical statistics does not have methods to analyze the problems in which random variables are vague (fuzzy). In this regard, a simple and new technique for testing the hypotheses under the fuzzy environments is proposed.  Here, the employed data are in terms of trapezoidal fuzzy numbers (TFN) which have been transformed into interval data using  interval method and on the grounds of the transformed fuzzy data, the one-factor ANOVA test is executed and decisions are concluded.  This concept has been illustrated by giving two numerical examples. Keywords: Fuzzy set, , Trapezoidal fuzzy number (TFN), Test of hypotheses, One-factor ANOVA model, Upper level data, Lower level data

Selection of Projects for Project Portfolio Using Fuzzy TOPSIS and Machine Learning

Project portfolio management (PPM) is extremely important nowadays due to the increasing severe competitions, accelerated product developments, project complexity, uncertainty, and challenges from global competitors. Therefore, businesses involved in many (dozens or even hundreds) projects need to formulate tactics and strategies to secure firms’ competencies and, most importantly, increase their productivities. Under this globalization context, PPM is to opti-mize the value provided to the customers while minimizing risks and the resources committed to the projects, while critical success factors (CSFs) is applied to anticipate the project’s risk and financial value by early assessment thus to help from the organizational level to predict the per-formance. Despite its importance, the literature on PPM and CSFs at a project level is rather limited, which demands a more profound knowledge about the assessment, ranking, and prior-itization of projects in the early stage. This study seeks to address the following two research questions: Do CSFs vary according to the project category, and how a supportive method can be established to help portfolio managers to select the project for portfolio. As a result, this re-search focuses on the multi-project context in order to fill the above-mentioned research gaps. As the contributions of this study, this study intends to (1) verify the hypothesis that different project category has different CSFs, and (2) contribute to explore how machine learning technol-ogy can be utilized for project selection. Projektisalkun hallinta (PPM) on nykyään erittäin tärkeää lisääntyvien kovien kilpailujen, nopeutuneen tuotekehityksen, projektien monimutkaisuuden, epävarmuuden ja globaalien kilpailijoiden haasteiden vuoksi. Siksi moniin (kymmeniin tai jopa satoihin) hankkeisiin osallistuvien yritysten on laadittava taktiikat ja strategiat, joilla varmistetaan yritysten osaaminen ja mikä tärkeintä, lisää tuottavuuttaan. Tässä globalisaatiokehyksessä PPM: n on optimoitava asiakkaille tarjottu arvo minimoiden riskit ja hankkeisiin sitoutuvat resurssit, kun taas kriittisiä menestystekijöitä (CSF) käytetään ennakoimaan projektin riski ja taloudellinen arvo varhaisella arvioinnilla, jotta apua organisaatiotasolta suorituskyvyn ennustamiseksi. Tärkeydestään huolimatta kirjallisuus PPM: stä ja CSF: stä projektitasolla on melko rajallinen, mikä vaatii syvällisempää tietoa hankkeiden arvioinnista, luokittelusta ja ennakoinnista varhaisessa vaiheessa. Tässä tutkimuksessa pyritään käsittelemään kahta seuraavaa tutkimuskysymystä: vaihtelevatko CSF: t projektikategorian mukaan ja kuinka voidaan luoda tukeva menetelmä salkunhoitajien auttamiseksi valitsemaan projekti salkkuun. Tämän seurauksena tämä uudelleenhaku keskittyy moniprojektiyhteyteen edellä mainittujen tutkimuksen aukkojen täyttämiseksi. Tämän tutkimuksen myötä tämän tutkimuksen tarkoituksena on (1) tarkistaa hypoteesi, että eri projektikategorioilla on erilaiset CSF: t, ja (2) myötävaikuttaa siihen, kuinka koneoppimisen tekniikkaa voidaan hyödyntää projektin valinnassa

Soft computing approaches to uncertainty propagation in environmental risk mangement

Real-world problems, especially those that involve natural systems, are complex and composed of many nondeterministic components having non-linear coupling. It turns out that in dealing with such systems, one has to face a high degree of uncertainty and tolerate imprecision. Classical system models based on numerical analysis, crisp logic or binary logic have characteristics of precision and categoricity and classified as hard computing approach. In contrast soft computing approaches like probabilistic reasoning, fuzzy logic, artificial neural nets etc have characteristics of approximation and dispositionality. Although in hard computing, imprecision and uncertainty are undesirable properties, in soft computing the tolerance for imprecision and uncertainty is exploited to achieve tractability, lower cost of computation, effective communication and high Machine Intelligence Quotient (MIQ). Proposed thesis has tried to explore use of different soft computing approaches to handle uncertainty in environmental risk management. The work has been divided into three parts consisting five papers. In the first part of this thesis different uncertainty propagation methods have been investigated. The first methodology is generalized fuzzy α-cut based on the concept of transformation method. A case study of uncertainty analysis of pollutant transport in the subsurface has been used to show the utility of this approach. This approach shows superiority over conventional methods of uncertainty modelling. A Second method is proposed to manage uncertainty and variability together in risk models. The new hybrid approach combining probabilistic and fuzzy set theory is called Fuzzy Latin Hypercube Sampling (FLHS). An important property of this method is its ability to separate randomness and imprecision to increase the quality of information. A fuzzified statistical summary of the model results gives indices of sensitivity and uncertainty that relate the effects of variability and uncertainty of input variables to model predictions. The feasibility of the method is validated to analyze total variance in the calculation of incremental lifetime risks due to polychlorinated dibenzo-p-dioxins and dibenzofurans (PCDD/F) for the residents living in the surroundings of a municipal solid waste incinerator (MSWI) in Basque Country, Spain. The second part of this thesis deals with the use of artificial intelligence technique for generating environmental indices. The first paper focused on the development of a Hazzard Index (HI) using persistence, bioaccumulation and toxicity properties of a large number of organic and inorganic pollutants. For deriving this index, Self-Organizing Maps (SOM) has been used which provided a hazard ranking for each compound. Subsequently, an Integral Risk Index was developed taking into account the HI and the concentrations of all pollutants in soil samples collected in the target area. Finally, a risk map was elaborated by representing the spatial distribution of the Integral Risk Index with a Geographic Information System (GIS). The second paper is an improvement of the first work. New approach called Neuro-Probabilistic HI was developed by combining SOM and Monte-Carlo analysis. It considers uncertainty associated with contaminants characteristic values. This new index seems to be an adequate tool to be taken into account in risk assessment processes. In both study, the methods have been validated through its implementation in the industrial chemical / petrochemical area of Tarragona. The third part of this thesis deals with decision-making framework for environmental risk management. In this study, an integrated fuzzy relation analysis (IFRA) model is proposed for risk assessment involving multiple criteria. The fuzzy risk-analysis model is proposed to comprehensively evaluate all risks associated with contaminated systems resulting from more than one toxic chemical. The model is an integrated view on uncertainty techniques based on multi-valued mappings, fuzzy relations and fuzzy analytical hierarchical process. Integration of system simulation and risk analysis using fuzzy approach allowed to incorporate system modelling uncertainty and subjective risk criteria. In this study, it has been shown that a broad integration of fuzzy system simulation and fuzzy risk analysis is possible. In conclusion, this study has broadly demonstrated the usefulness of soft computing approaches in environmental risk analysis. The proposed methods could significantly advance practice of risk analysis by effectively addressing critical issues of uncertainty propagation problem.Los problemas del mundo real, especialmente aquellos que implican sistemas naturales, son complejos y se componen de muchos componentes indeterminados, que muestran en muchos casos una relación no lineal. Los modelos convencionales basados en técnicas analíticas que se utilizan actualmente para conocer y predecir el comportamiento de dichos sistemas pueden ser muy complicados e inflexibles cuando se quiere hacer frente a la imprecisión y la complejidad del sistema en un mundo real. El tratamiento de dichos sistemas, supone el enfrentarse a un elevado nivel de incertidumbre así como considerar la imprecisión. Los modelos clásicos basados en análisis numéricos, lógica de valores exactos o binarios, se caracterizan por su precisión y categorización y son clasificados como una aproximación al hard computing. Por el contrario, el soft computing tal como la lógica de razonamiento probabilístico, las redes neuronales artificiales, etc., tienen la característica de aproximación y disponibilidad. Aunque en la hard computing, la imprecisión y la incertidumbre son propiedades no deseadas, en el soft computing la tolerancia en la imprecisión y la incerteza se aprovechan para alcanzar tratabilidad, bajos costes de computación, una comunicación efectiva y un elevado Machine Intelligence Quotient (MIQ). La tesis propuesta intenta explorar el uso de las diferentes aproximaciones en la informática blanda para manipular la incertidumbre en la gestión del riesgo medioambiental. El trabajo se ha dividido en tres secciones que forman parte de cinco artículos. En la primera parte de esta tesis, se han investigado diferentes métodos de propagación de la incertidumbre. El primer método es el generalizado fuzzy α-cut, el cual está basada en el método de transformación. Para demostrar la utilidad de esta aproximación, se ha utilizado un caso de estudio de análisis de incertidumbre en el transporte de la contaminación en suelo. Esta aproximación muestra una superioridad frente a los métodos convencionales de modelación de la incertidumbre. La segunda metodología propuesta trabaja conjuntamente la variabilidad y la incertidumbre en los modelos de evaluación de riesgo. Para ello, se ha elaborado una nueva aproximación híbrida denominada Fuzzy Latin Hypercube Sampling (FLHS), que combina los conjuntos de la teoría de probabilidad con la teoría de los conjuntos difusos. Una propiedad importante de esta teoría es su capacidad para separarse los aleatoriedad y imprecisión, lo que supone la obtención de una mayor calidad de la información. El resumen estadístico fuzzificado de los resultados del modelo generan índices de sensitividad e incertidumbre que relacionan los efectos de la variabilidad e incertidumbre de los parámetros de modelo con las predicciones de los modelos. La viabilidad del método se llevó a cabo mediante la aplicación de un caso a estudio donde se analizó la varianza total en la cálculo del incremento del riesgo sobre el tiempo de vida de los habitantes que habitan en los alrededores de una incineradora de residuos sólidos urbanos en Tarragona, España, debido a las emisiones de dioxinas y furanos (PCDD/Fs). La segunda parte de la tesis consistió en la utilización de las técnicas de la inteligencia artificial para la generación de índices medioambientales. En el primer artículo se desarrolló un Índice de Peligrosidad a partir de los valores de persistencia, bioacumulación y toxicidad de un elevado número de contaminantes orgánicos e inorgánicos. Para su elaboración, se utilizaron los Mapas de Auto-Organizativos (SOM), que proporcionaron un ranking de peligrosidad para cada compuesto. A continuación, se elaboró un Índice de Riesgo Integral teniendo en cuenta el Índice de peligrosidad y las concentraciones de cada uno de los contaminantes en las muestras de suelo recogidas en la zona de estudio. Finalmente, se elaboró un mapa de la distribución espacial del Índice de Riesgo Integral mediante la representación en un Sistema de Información Geográfico (SIG). El segundo artículo es un mejoramiento del primer trabajo. En este estudio, se creó un método híbrido de los Mapas Auto-organizativos con los métodos probabilísticos, obteniéndose de esta forma un Índice de Riesgo Integrado. Mediante la combinación de SOM y el análisis de Monte-Carlo se desarrolló una nueva aproximación llamada Índice de Peligrosidad Neuro-Probabilística. Este nuevo índice es una herramienta adecuada para ser utilizada en los procesos de análisis. En ambos artículos, la viabilidad de los métodos han sido validados a través de su aplicación en el área de la industria química y petroquímica de Tarragona (Cataluña, España). El tercer apartado de esta tesis está enfocado en la elaboración de una estructura metodológica de un sistema de ayuda en la toma de decisiones para la gestión del riesgo medioambiental. En este estudio, se presenta un modelo integrado de análisis de fuzzy (IFRA) para la evaluación del riesgo cuyo resultado depende de múltiples criterios. El modelo es una visión integrada de las técnicas de incertidumbre basadas en diseños de valoraciones múltiples, relaciones fuzzy y procesos analíticos jerárquicos inciertos. La integración de la simulación del sistema y el análisis del riesgo utilizando aproximaciones inciertas permitieron incorporar la incertidumbre procedente del modelo junto con la incertidumbre procedente de la subjetividad de los criterios. En este estudio, se ha demostrado que es posible crear una amplia integración entre la simulación de un sistema incierto y de un análisis de riesgo incierto. En conclusión, este trabajo demuestra ampliamente la utilidad de aproximación Soft Computing en el análisis de riesgos ambientales. Los métodos propuestos podría avanzar significativamente la práctica de análisis de riesgos de abordar eficazmente el problema de propagación de incertidumbre

The application of z-numbers in fuzzy decision making: The state of the art

A Z-number is very powerful in describing imperfect information, in which fuzzy numbers are paired such that the partially reliable information is properly processed. During a decision-making process, human beings always use natural language to describe their preferences, and the decision information is usually imprecise and partially reliable. The nature of the Z-number, which is composed of the restriction and reliability components, has made it a powerful tool for depicting certain decision information. Its strengths and advantages have attracted many researchers worldwide to further study and extend its theory and applications. The current research trend on Z-numbers has shown an increasing interest among researchers in the fuzzy set theory, especially its application to decision making. This paper reviews the application of Z-numbers in decision making, in which previous decision-making models based on Z-numbers are analyzed to identify their strengths and contributions. The decision making based on Z-numbers improves the reliability of the decision information and makes it more meaningful. Another scope that is closely related to decision making, namely, the ranking of Z-numbers, is also reviewed. Then, the evaluative analysis of the Z-numbers is conducted to evaluate the performance of Z-numbers in decision making. Future directions and recommendations on the applications of Z-numbers in decision making are provided at the end of this review