This document summarizes a study that used the fuzzy TOPSIS method to select the optimal type of spillway for a dam in northern Greece called Pigi Dam. Five alternative spillway types were evaluated based on nine criteria. The criteria were expressed as triangular fuzzy numbers to account for uncertainty. Weights for the criteria were determined using the AHP method and also expressed linguistically as fuzzy numbers. The fuzzy TOPSIS method was then used to rank the alternatives based on their distances from the ideal and negative-ideal solutions. The alternative with the highest relative closeness to the ideal solution was determined to be the optimal spillway type.
Ridge regression method is an improved method when the assumptions of independence of the explanatory variables cannot be achieved, which is also called multicollinearity problem, in regression analysis. One of the way to eliminate the multicollinearity problem is to ignore the unbiased property of . Ridge regression estimates the regression coefficients biased in order to decrease the variance of the regression coefficients.
A Moment Inequality for Overall Decreasing Life Class of Life Distributions w...inventionjournals
:A moment inequality is derived for the system whose life distribution is in an overall decreasing life (ODL) class of life distributions. A new nonparametric test statistic for testing exponentiality against ODL is investigated based on this inequality. The asymptotic normality of the proposed statistic is presented. Pitman's asymptotic efficiency, power and critical values of this test are calculated to assess the performance of the test. Real examples are given to elucidate the use of the proposed test statistic in the reliability analysis. Wealso proposed a test for testing exponentiality versus ODL for right censored data and the power estimates of this test are also simulated for censored data for some commonly used distributions in reliability. Finally, real data are used as an example for practical problems.
Ridge regression method is an improved method when the assumptions of independence of the explanatory variables cannot be achieved, which is also called multicollinearity problem, in regression analysis. One of the way to eliminate the multicollinearity problem is to ignore the unbiased property of . Ridge regression estimates the regression coefficients biased in order to decrease the variance of the regression coefficients.
A Moment Inequality for Overall Decreasing Life Class of Life Distributions w...inventionjournals
:A moment inequality is derived for the system whose life distribution is in an overall decreasing life (ODL) class of life distributions. A new nonparametric test statistic for testing exponentiality against ODL is investigated based on this inequality. The asymptotic normality of the proposed statistic is presented. Pitman's asymptotic efficiency, power and critical values of this test are calculated to assess the performance of the test. Real examples are given to elucidate the use of the proposed test statistic in the reliability analysis. Wealso proposed a test for testing exponentiality versus ODL for right censored data and the power estimates of this test are also simulated for censored data for some commonly used distributions in reliability. Finally, real data are used as an example for practical problems.
Multiple Linear Regression Model with Two Parameter Doubly Truncated New Symm...theijes
The most commonly used method to describe the relationship between response and independent variables is a linear model with Gaussian distributed errors. In practical components, the variables examined might not be mesokurtic and the populace values probably finitely limited. In this paper, we introduce a multiple linear regression models with two-parameter doubly truncated new symmetric distributed (DTNSD) errors for the first time. To estimate the model parameters we used the method of maximum likelihood (ML) and ordinary least squares (OLS). The model desires criteria such as Akaike information criteria (AIC) and Bayesian information criteria (BIC) for the models are used. A simulation study is performed to analysis the properties of the model parameters. A comparative study of doubly truncated new symmetric linear regression models on the Gaussian model showed that the proposed model gives good fit to the data sets for the error term follow DTNSD
Comparison of Max100, SWARA and Pairwise Weight Elicitation MethodsIJERA Editor
Decision making is used in every part of life and realised by each action taken. The presence of correct and satisfactory solution to problems is very important for person, institution and organizations. Multiple Criteria Decision Making (MCDM) techniques are developed for this purpose. Based upon the former studies, it is seen that weight elicitation methods used in solving MCDM problems, have an important role at defining the importance of criteria and obtaining the best and satisfying results for decision makers. Theaim of the paperis to compare the results of range variability between the criteria for Max100, Stepwise Weight Assessment Ratio Analysis (SWARA) and Pairwise Comparison weight elicitation methods and to give suggestion about conditions of using of the methods. It is the first time SWARA is compared with Pairwise Comparison and Max100 methods, and it makes this study different. When results of the study is considered, it is seen that variability of Pairwise Comparison method is higher than that Max100 and SWARA methods. Besides, Max100 is found as the easiest method to use, and Pairwise Comparison method’s way of scoring is defined as the most reliable. In the light of the results obtained from the methods, some conditions of usage are suggested.
Hellinger Optimal Criterion and 퓗푷푨- Optimum Designs for Model Discrimination...inventionjournals
Kullback-Leibler (KL) optimality criterion has been considered in the literature for model discrimination. However, Hellinger distance has many advantages rather than KL-distance. For that reason, in this paper a new criterion based on the Hellinger distance named by Hellinger (ℋ) -optimality criterion is proposed to discriminate between two rival models. An equivalence theorem is proved for this criterion. Furthermore, a new compound criterion is constructed that possess both discrimination and a high probability of desired outcome properties. Discrimination between binary and Logistic GLM are suggested based on the new criteria
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Please Subscribe to this Channel for more solutions and lectures
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Elementary Statistics Practice Test 1
Module 1: Chapters 1-3
Chapter 1: Introduction to Statistics.
Chapter 2: Exploring Data with Tables and Graphs.
Chapter 3: Describing, Exploring, and Comparing Data.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 6: Normal Probability Distribution
6.1: The Standard Normal Distribution
Modeling monthly average daily diffuse radiation for dhaka, bangladesheSAT Journals
Abstract The diffuse part of solar radiation is one of the elements necessary for the design and evaluation of energy production of a solar system. However, in most cases, when radiometric measurements are made, only global radiation is available. To remedy this situation, this paper presents a model of the scattered radiation measured on a horizontal surface for the capital city of Bangladesh. The correlation established for the chosen site was compared to the work of Liu anf Jordan, Page, Collares Pereira and Rabl, Modi and Sukhatme and Gupta el al. Keywords: Diffuse Radiation, Clearness Index, Regression analysis, Horizontal Radiation.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
Study of Silica Fume Blended Concrete using Statistical Mixture Experiment an...ijtsrd
The prime objective of this study is to design, analyze and optimize the silica fume blended concrete using Statistical Mixture Experiment and to explore the practicability of Statistical mixture experiment in mix design of concrete containing the several components. The traditional mix design methods have drawbacks. Traditional method of mix design of concrete fails in designing the complex concrete mixture containing the cementitious materials, pozzolanic materials and several admixtures. Traditional methods are based on trial and error method in which effect of only one component can be determined. Trial and error does not provide the whole picture of concrete mixture. In this study, the statistical mixture design method is employed to analyze the effects of silica fume on 28 days compressive strength of concrete and to design and optimize the silica fume blended concrete. The components of concrete mixture are Cement, silica fume, water and aggregate. Total of 15 runs or design points are generated which are combinations of proportions and laboratory experiment is carried out on these design points and 28 days compressive strength is measured. Mathematical relationship is established between 28 days compressive strength and proportions of components using least square technique and then mathematical model is studied to analyze the effects of silica fume in concrete. Pankaj Kumar | M. C Paliwal ""Study of Silica Fume Blended Concrete using Statistical Mixture Experiment and Optimizing of Compressive Strength of Cement Concrete"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-4 , June 2019, URL: https://www.ijtsrd.com/papers/ijtsrd25178.pdf
Paper URL: https://www.ijtsrd.com/engineering/civil-engineering/25178/study-of-silica-fume-blended-concrete-using-statistical-mixture-experiment-and-optimizing-of-compressive-strength-of-cement-concrete/pankaj-kumar
Many Decision Problems in business and social systems can be modeled using mathematical optimization, which seeks to maximize or minimize some objective which is a function of the decisions.
Stochastic Optimization Problems are mathematical programs where some of the data incorporated into the objective or constraints are Uncertain.
whereas, Deterministic Optimization Problems are formulated with known parameters.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Integrating Fuzzy Dematel and SMAA-2 for Maintenance Expensesinventionjournals
: The majority of the allowances being transferred to public institutions are mostly spent for buying new equipment, materials, facilities and their maintenance and repair. Some of the public sectors establish their own plants in order to reduce the maintenance and repair costs and gain ability to perform these activities. However, developing technology and variety of materials make their repair and maintenance activities more expensive for them. In this study, vital criteria for a public institution are determined. By using Fuzzy DEMATEL (Decision Making Trial And Evaluation Laboratory) method the degree of importance is identified by two defuzzification methods and the alternatives are ranked by using SMAA-2 (Stochastic Multi Criteria Acceptability Analysis) in three scenarios. The results show that different defuzzification methods change the order of preferences.
Multiple Linear Regression Model with Two Parameter Doubly Truncated New Symm...theijes
The most commonly used method to describe the relationship between response and independent variables is a linear model with Gaussian distributed errors. In practical components, the variables examined might not be mesokurtic and the populace values probably finitely limited. In this paper, we introduce a multiple linear regression models with two-parameter doubly truncated new symmetric distributed (DTNSD) errors for the first time. To estimate the model parameters we used the method of maximum likelihood (ML) and ordinary least squares (OLS). The model desires criteria such as Akaike information criteria (AIC) and Bayesian information criteria (BIC) for the models are used. A simulation study is performed to analysis the properties of the model parameters. A comparative study of doubly truncated new symmetric linear regression models on the Gaussian model showed that the proposed model gives good fit to the data sets for the error term follow DTNSD
Comparison of Max100, SWARA and Pairwise Weight Elicitation MethodsIJERA Editor
Decision making is used in every part of life and realised by each action taken. The presence of correct and satisfactory solution to problems is very important for person, institution and organizations. Multiple Criteria Decision Making (MCDM) techniques are developed for this purpose. Based upon the former studies, it is seen that weight elicitation methods used in solving MCDM problems, have an important role at defining the importance of criteria and obtaining the best and satisfying results for decision makers. Theaim of the paperis to compare the results of range variability between the criteria for Max100, Stepwise Weight Assessment Ratio Analysis (SWARA) and Pairwise Comparison weight elicitation methods and to give suggestion about conditions of using of the methods. It is the first time SWARA is compared with Pairwise Comparison and Max100 methods, and it makes this study different. When results of the study is considered, it is seen that variability of Pairwise Comparison method is higher than that Max100 and SWARA methods. Besides, Max100 is found as the easiest method to use, and Pairwise Comparison method’s way of scoring is defined as the most reliable. In the light of the results obtained from the methods, some conditions of usage are suggested.
Hellinger Optimal Criterion and 퓗푷푨- Optimum Designs for Model Discrimination...inventionjournals
Kullback-Leibler (KL) optimality criterion has been considered in the literature for model discrimination. However, Hellinger distance has many advantages rather than KL-distance. For that reason, in this paper a new criterion based on the Hellinger distance named by Hellinger (ℋ) -optimality criterion is proposed to discriminate between two rival models. An equivalence theorem is proved for this criterion. Furthermore, a new compound criterion is constructed that possess both discrimination and a high probability of desired outcome properties. Discrimination between binary and Logistic GLM are suggested based on the new criteria
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 1
Module 1: Chapters 1-3
Chapter 1: Introduction to Statistics.
Chapter 2: Exploring Data with Tables and Graphs.
Chapter 3: Describing, Exploring, and Comparing Data.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 6: Normal Probability Distribution
6.1: The Standard Normal Distribution
Modeling monthly average daily diffuse radiation for dhaka, bangladesheSAT Journals
Abstract The diffuse part of solar radiation is one of the elements necessary for the design and evaluation of energy production of a solar system. However, in most cases, when radiometric measurements are made, only global radiation is available. To remedy this situation, this paper presents a model of the scattered radiation measured on a horizontal surface for the capital city of Bangladesh. The correlation established for the chosen site was compared to the work of Liu anf Jordan, Page, Collares Pereira and Rabl, Modi and Sukhatme and Gupta el al. Keywords: Diffuse Radiation, Clearness Index, Regression analysis, Horizontal Radiation.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
Study of Silica Fume Blended Concrete using Statistical Mixture Experiment an...ijtsrd
The prime objective of this study is to design, analyze and optimize the silica fume blended concrete using Statistical Mixture Experiment and to explore the practicability of Statistical mixture experiment in mix design of concrete containing the several components. The traditional mix design methods have drawbacks. Traditional method of mix design of concrete fails in designing the complex concrete mixture containing the cementitious materials, pozzolanic materials and several admixtures. Traditional methods are based on trial and error method in which effect of only one component can be determined. Trial and error does not provide the whole picture of concrete mixture. In this study, the statistical mixture design method is employed to analyze the effects of silica fume on 28 days compressive strength of concrete and to design and optimize the silica fume blended concrete. The components of concrete mixture are Cement, silica fume, water and aggregate. Total of 15 runs or design points are generated which are combinations of proportions and laboratory experiment is carried out on these design points and 28 days compressive strength is measured. Mathematical relationship is established between 28 days compressive strength and proportions of components using least square technique and then mathematical model is studied to analyze the effects of silica fume in concrete. Pankaj Kumar | M. C Paliwal ""Study of Silica Fume Blended Concrete using Statistical Mixture Experiment and Optimizing of Compressive Strength of Cement Concrete"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-4 , June 2019, URL: https://www.ijtsrd.com/papers/ijtsrd25178.pdf
Paper URL: https://www.ijtsrd.com/engineering/civil-engineering/25178/study-of-silica-fume-blended-concrete-using-statistical-mixture-experiment-and-optimizing-of-compressive-strength-of-cement-concrete/pankaj-kumar
Many Decision Problems in business and social systems can be modeled using mathematical optimization, which seeks to maximize or minimize some objective which is a function of the decisions.
Stochastic Optimization Problems are mathematical programs where some of the data incorporated into the objective or constraints are Uncertain.
whereas, Deterministic Optimization Problems are formulated with known parameters.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Integrating Fuzzy Dematel and SMAA-2 for Maintenance Expensesinventionjournals
: The majority of the allowances being transferred to public institutions are mostly spent for buying new equipment, materials, facilities and their maintenance and repair. Some of the public sectors establish their own plants in order to reduce the maintenance and repair costs and gain ability to perform these activities. However, developing technology and variety of materials make their repair and maintenance activities more expensive for them. In this study, vital criteria for a public institution are determined. By using Fuzzy DEMATEL (Decision Making Trial And Evaluation Laboratory) method the degree of importance is identified by two defuzzification methods and the alternatives are ranked by using SMAA-2 (Stochastic Multi Criteria Acceptability Analysis) in three scenarios. The results show that different defuzzification methods change the order of preferences.
Integrating Fuzzy Dematel and SMAA-2 for Maintenance Expensesinventionjournals
The majority of the allowances being transferred to public institutions are mostly spent for buying new equipment, materials, facilities and their maintenance and repair. Some of the public sectors establish their own plants in order to reduce the maintenance and repair costs and gain ability to perform these activities. However, developing technology and variety of materials make their repair and maintenance activities more expensive for them. In this study, vital criteria for a public institution are determined. By using Fuzzy DEMATEL (Decision Making Trial And Evaluation Laboratory) method the degree of importance is identified by two defuzzification methods and the alternatives are ranked by using SMAA-2 (Stochastic Multi Criteria Acceptability Analysis) in three scenarios. The results show that different defuzzification methods change the order of preferences.
RESIDUALS AND INFLUENCE IN NONLINEAR REGRESSION FOR REPEATED MEASUREMENT DATAorajjournal
All observations don’t have equal significance in regression analysis. Diagnostics of observations is an important aspect of model building. In this paper, we use diagnostics method to detect residuals and influential points in nonlinear regression for repeated measurement data. Cook distance and Gauss newton method have been proposed to identify the outliers in nonlinear regression analysis and parameter estimation. Most of these techniques based on graphical representations of residuals, hat matrix and case deletion measures. The results
show us detection of single and multiple outliers cases in repeated measurement data. We use these techniques
to explore performance of residuals and influence in nonlinear regression model.
Researchers use several tools and procedures for analyzing quantitative data obtained from different types of experimental designs. Different designs call for different methods of analysis. This presentation focuses on:
T-test
Analysis of variance (F-test), and
Chi-square test
Refining Measure of Central Tendency and DispersionIOSR Journals
A unified approach is attempted to bring the descriptive statistics in to a more refined frame work. Different measure of central tendencies such as arithmetic mean, median, mode, geometric mean and harmonic mean are derived from a generalized notion of a measure of central tendency developed through an optimality criteria. This generalized notion is extended to introduce the concept of an interval of central tendency. Retaining the spirit of this notion, measure of central tendency may be called point of central tendency. The same notion is further extended to obtain confidence interval for population mean in a finite population model and confidence interval for probability of success in Bernoulli population.
In recent years, consumers and legislation have been pushing companies to optimize their activities in such a way as to reduce negative environmental and social impacts more and more. In the other side, companies
must keep their total supply chain costs as low as possible to remain competitive.This work aims to develop a model to traveling salesman problem including environmental impacts and to identify, as far as possible, the contribution of genetic operator’s tuning and setting in the success and
efficiency of genetic algorithms for solving this problem with consideration of CO2 emission due to transport. This efficiency is calculated in terms of CPU time consumption and convergence of the solution. The best transportation policy is determined by finding a balance between financial and environmental
criteria.Empirically, we have demonstrated that the performance of the genetic algorithm undergo relevant
improvements during some combinations of parameters and operators which we present in our results part.
The utilization of grey sets theory in multi metrics modeling to benchmark sustainability of system’s organizations under integrated approach helps for decision making. EWZ (Environmental, Minimization of the Waste and Zero Defect) measurement of each production unit simultaneously is considered complex problem. In the presented research work, the authors have proposed a DSS, consist of implementation Grey-FMF FMF (Full Multification Form) upon EWZ ‘Environmental, Minimization of the Waste and Zero Defect’ system sustainability model to evaluate the performance‘s score of system’s organization under partial information of team of professionals (Ps) against vague practices.
Dimensionality Reduction Techniques In Response Surface Designsinventionjournals
Dimensionality reduction has enormous applications in various fields in industries. It can be applied in an optimal way with respect to time and cost related to the agricultural sciences, mechanical engineering, chemical technology, pharmaceutical sciences, clinical trials, biological studies, image processing, pattern recognitions etc. Several researchers made attempts on the reduction of the size of the model for different specific problems using some mathematical and statistical techniques identifying and eliminating some insignificant variables.This paper presents a review of the available literature on dimensionality reduction
Grey Multi Criteria Decision Making MethodsIJSRP Journal
Multi-Criteria Decision Making is the most well-known branch of decision making. In some cases, determining precisely the exact value of attributes is difficult and their values can be considered as Uncertain data. This paper presents two different Multi Criteria Decision Making methods based on grey numbers. The two methods are used to obtain the final ranking of the alternatives and select the best one under grey numbers. Finally, an illustrative example is presented and the results are analyzed.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
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Biological screening of herbal drugs: Introduction and Need for
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Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
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Antifertility, Toxicity studies as per OECD guidelines
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
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1. Proceedings 2018, 2, 637; doi:10.3390/proceedings2110637 www.mdpi.com/journal/proceedings
Proceedings
Multi-Criteria Decision Making Using TOPSIS
Method Under Fuzzy Environment. Application in
Spillway Selection †
Vasiliki Balioti *, Christos Tzimopoulos and Christos Evangelides
Department of Transportation and Hydraulic Engineering, School of Rural & Surveying Engineering, Faculty
of Engineering, Aristotle University, 54124 Thessaloniki, Greece; tzimop@eng.auth.gr (C.T.);
evan@eng.auth.gr (C.E.)
* Correspondence: vasilikimpalioti@hotmail.com; Tel.: +30-693-932-0723
† Presented at the 3rd EWaS International Conference on “Insights on the Water-Energy-Food Nexus”,
Lefkada Island, Greece, 27–30 June 2018.
Published: 31 July 2018
Abstract: The selection of an appropriate spillway has a significant effect to the construction of a
dam and several procedures and considerations are needed. In the past, this selection of the type of
the spillway was arbitrary and sometimes with bad results. Recently the Multiple Criteria Decision
Making theory has given the possibility to make a decision about the optimum form of a spillway
under complex circumstances. In this paper, the above method is used and especially the TOPSIS
(Technique for Order Preference by Similarity to Ideal Solution) method for the selection of a
spillway for a dam in the district of Kilkis in Northern Greece—‘Dam Pigi’. As the criteria were
fuzzy and uncertain, the Fuzzy TOPSIS method is introduced together with the AHP (Analytic
Hierarchy Process), which is used for the evaluation of criteria and weights. Five types of spillways
were selected as alternatives and nine criteria. The criteria are expressed as triangular fuzzy
numbers in order to formulate the problem. Finally, using the Fuzzy TOPSIS method, the
alternatives were ranked and the optimum type of spillway was obtained.
Keywords: spillway selection; fuzzy TOPSIS method; MCDM; AHP method
1. Introduction
The optimal type of a spillway is one of the most complex issues in water management including
fuzziness due to the existence of qualitative criteria and the uncertainty in evaluating them. Every
spillway presents some advantages and disadvantages, technical, financial, environmental etc. which
set a group of constraints. Therefore a comparative evaluation is needed to reach a scientific and
sufficiently justified solution.
Multiple criteria decision-making (MCDM) is considered as a sophisticated decision-making
tool involving both quantitative and qualitative factors. In recent years, several MCDM techniques
and approaches have been suggested in order to choose the probable optimal options. An extension
to the fuzzy multiple criteria decision making (MCDM) model is suggested in this work, where the
ratings of alternatives versus criteria, and the importance weights of all criteria, are assessed in
linguistic values represented by fuzzy numbers.
Specifically, an extension of the TOPSIS method in a fuzzy environment is adopted. Many
researchers have developed the model of similarity to ideal solution to the fuzzy environment and
have utilized it in various fields. Chen [1] expanded the TOPSIS method for decision-making
problems to the fuzzy environment. According to this theory, the attributes are expressed in TFNs
(Triangular Fuzzy Numbers), the normalization method is linear and vertex method is proposed for
2. Proceedings 2018, 2, 637 2 of 8
the calculation of the distance measurements for the final ranking. In parallel, a second normalization
technique is utilized from the article of Jahanshahloo et al. [2].
This paper is one of the first applications of the fuzzy TOPSIS method in solving water
management or hydraulic problems.
2. Methods
2.1. TOPSIS (Technique for Order Preference by Similarity to Ideal Solution)
The TOPSIS method was first developed in 1981 by Yoon and Hwang [3]. Its basic concept is
that the chosen alternative should have the shortest distance from the ideal solution and the farthest
from the negative-ideal solution (Figure 1) [3,4].
Figure 1. Basic concept of TOPSIS method (A+: Ideal point, A−: Negative—Ideal Point).
According to Chen’s approach [1], the procedure of fuzzy TOPSIS is similar to the classic one
and can be expressed in a series of steps:
1. Construct the normalized decision matrix.
In the fuzzy environment, in order to avoid the complicated normalization formula used in
classical TOPSIS, simpler formulas are used to transform the various criteria scales into a comparable
scale.
• The linear scale transformation [1] is:
= *
j
ij
*
j
ij
*
j
ij
ij
c
c
,
c
b
,
c
a
r
~ , ij
i
*
j c
max
c = (1)
• Jahanshahloo et al. formula [2] is:
( ) ( )
=
+
=
=
+
=
n
1
i
2
)
ij
(c
2
)
ij
(a
ij
c
,
n
1
i ij
2b
ij
b
,
n
1
i
2
)
ij
(c
2
)
ij
(a
ij
a
ij
r
~ (2)
where ( )
ij
ij
ij
ij c
,
b
,
a
x
~ = are the elements of the decision matrix.
2. Construct the weighted normalized decision matrix.
ij
j
ij r
~
w
~
ν
~ ⋅
= , j = 1, 2, … m, i = 1, 2, … n (3)
3. Determine the fuzzy ideal and fuzzy negative-ideal solutions.
3. Proceedings 2018, 2, 637 3 of 8
}
ν
~
,...
ν
~
,
ν
~
{
A m
2
1
+
+
+
+
= (4)
}
ν
~
,...
ν
~
,
ν
~
{
A m
2
1
−
−
−
−
= (5)
where +
j
ν
~ = (1,1,1) and −
j
ν
~ = (0,0,0), j = 1, 2, … m.
4. Calculate the separation measure:
• Ideal separation
=
+
+
=
m
1
j
j
ij
i )
ν
~
,
ν
~
s(
S i = 1, 2, … n (6)
• Negative-ideal separation
=
−
−
=
m
1
j
j
ij
i )
ν
~
,
ν
~
s(
S i = 1, 2, … n (7)
where )
ν
~
,
ν
~
s( j
ij
+
and )
ν
~
,
ν
~
s( j
ij
−
are distance measurements calculated with the vertex method:
d( ij
x
~ , ij
y
~ ) = ( ) ( ) ( )
−
+
−
+
−
2
3
ij
y
3
ij
x
2
2
ij
y
2
ij
x
2
1
ij
y
1
ij
x
3
1
(8)
( )
3
ij
x
,
2
ij
x
,
1
ij
x
ij
x
~ = , ( )
3
ij
y
,
2
ij
y
,
1
ij
y
ij
y
~ =
5. Calculate the relative closeness to the Ideal Solution.
)
S
(S
S
c
i
i
i
*
i −
+
−
+
= , 0 <
*
i
c < 1, i = 1, 2, …, n (9)
*
i
c = 1 if Ai = A+
*
i
c = 0 if Ai = A−
6. Rank the preference order.
A set of alternatives can now be preference ranked according to the descending order of
*
i
c [3,4].
The method presupposes that [3,4]:
• Each criterion in the decision matrix takes either monotonically increasing or monotonically
decreasing utility.
• A decision matrix of n alternatives and m criteria and a set of weights for the criteria are required.
• Any outcome which is expressed in a non-numerical way should be quantified through the
appropriate scaling technique.
2.2. AHP (Analytic Hierarchy Process)
The analytic hierarchy process (AHP) [5] is based on decomposing a complex MCDM problem
into a system of hierarchies. There is a fundamental 1–9 scale of absolute numbers shown in Table 1,
in order to design the hierarchy.
When we use judgment to estimate dominance in making comparisons, and in particular when
the criterion of the comparisons is intangible, instead of using two numbers wi and wj from a scale
(rather than interpreting the significance of their ratio wi/wj) we assign a single number drawn from
the fundamental 1–9 scale of absolute numbers shown in Table 1 to represent the ratio (wi/wj)/1. It is
the nearest integer approximation to the ratio wi/wj. The derived scale will reveal what the wi and wj
4. Proceedings 2018, 2, 637 4 of 8
are. This is a central fact about the relative measurement approach and the need for a fundamental
scale.
Table 1. Fundamental Scale of Absolute Numbers.
Intensity of Importance Definition Explanation
1 Equal Importance
Two activities contribute equally to
the objective
3 Moderate importance
Experience and judgment slightly
favor one activity over another
5 Strong importance
Experience and judgment strongly
favor one activity over another
7 Very strong or demonstrated importance
An activity is favored very strongly
over another; its dominance
demonstrated in practice
9 Extreme importance
The evidence favoring one activity
over another is of the highest possible
order of affirmation
Reciprocals of above
If activity i has one of the above nonzero numbers
assigned to it when compared with activity j, then j has
the reciprocal value when compared with i
A logical assumption
Intensities of 2, 4, 6 and 8 can be used to express intermediate values.
2.3. Linguistic Variables
The extension of the TOPSIS method in the fuzzy environment can be achieved by expressing
the weights of the criteria and the ratings as linguistic variables. A linguistic variable is a variable
whose values are linguistic terms. The concept of linguistic variable is very useful in dealing with
situations which are too complex or too ill-defined to be reasonably described in conventional
quantitative expressions [6]. According to Chen, the linguistic variables can be expressed in positive
triangular fuzzy numbers as shown in Tables 2 and 3 [1].
Table 2. Linguistic variables for the importance weight of each criterion.
Very low (VL) (0.1, 0, 0)
Low (L) (0.3, 0.1, 0.1)
Medium low (ML) (0.5, 0.3, 0.3)
Medium (M) (0.7 ,0.5, 0.5)
Medium high (MH) (0.9, 0.7, 0.7)
High (H) (1, 0.9, 0.9)
Very high (VH) (1, 1, 1)
Table 3. Linguistic variables for the ratings.
Very poor (VP) (1, 0, 0)
Poor (P) (3, 1, 1)
Medium poor (MP) (5, 3, 3)
Fair (F) (7, 5, 5)
Medium good (MG) (9, 7, 7)
Good (G) (10, 9, 9)
Very good (VG) (10, 10, 10)
3. Illustrative application
3.1. General Information
The dam which was chosen for this application is “Pigi Dam” on the “Kotza-Dere” river and it
is located in the north of Greece. The dam is considered as a large dam and was constructed in 1999.
It is a rockfill dam constructed for irrigation. There are five alternative types of the spillway, based
5. Proceedings 2018, 2, 637 5 of 8
on their technical efficiency for the dam which was selected; (a) X1 = ogee or overfall spillway, (b) X2
= shaft or morning glory spillway, (c) X3 = side channel spillway, (d) X4 = siphon spillway and (e) X5
= gated spillway.
3.2. Criteria
The main criteria were determined using extensive library studies and experts’ opinion. The
institutes that were found to give special recommendations for the selection of the type of the
spillway are the Indian Standards Institute [7] and U.S. Bureau of Reclamation [8]. Finally, nine
criteria have been chosen: (a) C1 = construction costs, (b) C2 = maintenance costs, (c) C3 = foundation,
(d) C4 = reservoir capacity, (e) C5 = static/construction difficulty, (f) C6 = discharge capacity, (g) C7 =
physical space, (h) C8 = conveyance feature (costs and construction difficulty) and (i) C9 = aesthetic.
Firstly, the decision maker constructs the pair-wise comparison matrix of the criteria (Table 4).
Since the consistency ratio (C.R.) is less than 0.1 or close, the judgments are acceptable. Finally, in
order to calculate the weights of the criteria, according to Saaty, the eigenvector is calculated (Table
5). In Figure 2 the column chart, whose values derive from Table 5, shows the weights’ comparison
across the criteria.
Table 4. Comparison matrix of the weights.
C1 C2 C3 C4 C5 C6 C7 C8 C9
C1 1 1 4 4 4 5 7 3 9
C2 1 1 4 4 4 5 7 3 9
C3 0.25 0.25 1 0.2 1 4 6 0.25 6
C4 0.25 0.25 5 1 5 6 5 0.25 7
C5 0.25 0.25 1 0.2 1 5 5 0.25 6
C6 0.20 0.20 0.25 0.17 0.20 1 3 0.2 4
C7 0.14 0.14 0.17 0.20 0.20 0.33 1 0.14 5
C8 0.33 0.33 4 4 4 5 7 1 8
C9 0.11 0.11 0.17 0.14 0.17 0.25 0.20 0.13 1
Table 5. Eigenvector—Criteria’s weights.
C1 C2 C3 C4 C5 C6 C7 C8 C9
0.235 0.235 0.074 0.133 0.075 0.039 0.029 0.164 0.015
Subsequently, fuzziness is introduced to the process so as to confront the uncertainties of
judgments or calculations of the previous step. The decision maker in Table 6 reconstructs Table 5
using linguistic variables and corresponds them to the TFNs proposed by Chen in Tables 2 and 3.
Figure 2. Importance of each criterion after AHP evaluation.
C1 C2 C3 C4 C5 C6 C7 C8 C9
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Table 6. Linguistic and fuzzy weights.
C1 C2 C3 C4 C5 C6 C7 C8 C9
VH VH ML M ML L L MH L
(0.9, 1, 1) (0.9, 1, 1) (0.1, 0.3, 0.5) (0.3, 0.5, 0.7) (0.1, 0.3, 0.5) (0, 0.1, 0.3) (0, 0.1, 0.3) (0.5, 0.7, 0.9) (0, 0.1, 0.3)
3.3. Decision Matrix
Initially, the alternatives are compared with pair-wise comparisons for each criterion. The
eigenvectors, which are calculated for each criterion, form the columns of the decision matrix (Table
7). The process of determining the decision matrix expressed in linguistic terms (Table 8) and the
fuzzy decision matrix (Table 9) is similar to determining the matrix of fuzzy weights, described above.
Table 7. Decision matrix after AHP evaluation.
C1 C2 C3 C4 C5 C6 C7 C8 C9
X1 0.22 0.50 0.29 0.08 0.36 0.06 0.13 0.11 0.28
X2 0.47 0.13 0.06 0.08 0.04 0.52 0.34 0.04 0.52
X3 0.22 0.26 0.06 0.08 0.08 0.06 0.13 0.28 0.06
X4 0.05 0.07 0.29 0.08 0.16 0.28 0.34 0.28 0.06
X5 0.03 0.03 0.29 0.67 0.36 0.06 0.06 0.28 0.06
Table 8. Decision matrix in linguistic terms.
C1 C2 C3 C4 C5 C6 C7 C8 C9
X1 F VG VG P VG P MP F F
X2 VG MP P P P VG VG P VG
X3 F F P P MP P MP VG P
X4 P P VG P F F VG VG P
X5 P P VG VG VG P MP VG P
Table 9. Fuzzy decision matrix.
C1 C2 C3 C4 C5 C6 C7 C8 C10
Χ1 (3, 5, 7) (9, 10, 10) (9, 10, 10) (0, 1, 3) (9, 10, 10) (0, 1, 3) (1, 3, 5) (3, 5, 7) (3, 5, 7)
Χ2 (9, 10, 10) (1, 3, 5) (0, 1, 3) (0, 1, 3) (0, 1, 3) (9, 10, 10) (9, 10, 10) (0, 1, 3) (9, 10, 10)
Χ3 (3, 5, 7) (3, 5, 7) (0, 1, 3) (0, 1, 3) (1, 3, 5) (0, 1, 3) (1, 3, 5) (9, 10, 10) (0, 1, 3)
Χ4 (0, 1, 3) (0, 1, 3) (9, 10, 10) (0, 1, 3) (3, 5, 7) (3, 5, 7) (9, 10, 10) (9, 10, 10) (0, 1, 3)
Χ5 (0, 1, 3) (0, 1, 3) (9, 10, 10) (9, 10, 10) (9, 10, 10) (0, 1, 3) (1, 3, 5) (9, 10, 10) (0, 1, 3)
3.4. Results
The optimal type of spillway is obtained by completing the TOPSIS’ calculations (Equations (1)–
(9)). For the calculations needed, two programs were utilized in Visual Fortran [9]. The ranking with
the two normalization approaches are:
• The linear scale transformation [1]:
X1(0.313) > X5(0.261) > X3(0.246) > X2(0.243) > X4(0.224)
X1 = ogee spillway > X5 = gated spillway > X3 = side channel spillway > X2 = shaft spillway > X4 =
siphon spillway
• Jahanshahloo et al. formula [2]:
X1(0.171) > X2(0.139) > X5(0.137) > X3(0.130) > X4(0.111)
X1 = ogee spillway > X2 = shaft spillway > X5 = gated spillway > X3 = side channel spillway > X4 =
siphon spillway.
7. Proceedings 2018, 2, 637 7 of 8
In Figure 3, the charts present the results of calculating the separation measures, Si+ and Si−
(Equations (6) and (7)) with each of the two approaches.
(a) (b)
Figure 3. Separation measures; Ideal separation (Si+). Negative-ideal separation (Si−). (a) 1st approach
(linear scale transformation); (b) 2nd approach (Jahanshahloo et al. formula).
4. Discussion and Conclusions
This paper presents the first application of the combination of the proposed MCDM methods
with fuzzy logic to solve the problem of selecting the optimal type of a spillway.
Although a spillway is a significant structure, little guidelines can be found for this selection. As
a result, in most cases, the selection derives from a techno-economic feasibility and analysis. Our
investigation proves that the result should take into consideration more parameters than technical
feasibility and low construction costs. X2 (shaft or morning glory spillway) spillway has the highest
evaluation for C1 (construction costs) criterion (Table 8), which means that it is the most cost-efficient
alternative. Nevertheless, in the presented approaches, X2 would be the fourth or the second choice
out of the five depending on the normalization technique.
Although the two approaches do not lead to the same ranking, the optimal choice is Χ1, the ogee
spillway and the worst is X4, the siphon spillway. The difference in ranking could be justified as the
values of the relative closeness for the alternatives Χ2, X3 and X5 are very close to the second approach
and no clear order could be obtained.
It is also notable that the separation measures, ideal and negative-ideal separations, have the
same ranking as the final one of the relative closeness.
Finally, it is suggested that engineering problems involving decision making be dealt with
MCDM methods and fuzzy logic. The aforementioned process, with either approach, could also be
implemented in various multi-criteria engineering problems.
References
1. Chen, C.T. Extensions of the TOPSIS for group decision-making under fuzzy environment. Fuzzy Sets Syst.
2000, 114, 1–9.
2. Jahanshahloo, G.R.; Lotfi, F.H.; Izadikhah, M. Extension of the TOPSIS method for decision-making
problems with fuzzy data. Appl. Math. Comput. 2006, 181, 1544–1551.
3. Yoon, K.; Hwang, C.L. Multiple Attribute Decision Making: Methods and Applications; Springer:
Berlin/Heidelberg, Germany, 1981.
4. Yoon, K. System Selection by Multiple Attribute Decision Making. Ph.D. Dissertation, Kansas State
University, Manhattan, KS, USA, 1980.
6.49 7.16 7.07 7.35 6.98
2.96 2.29 2.31 2.12 2.47
0
2
4
6
8
10
X1 X2 X3 X4 X5
Separation
measures
Spillway
Si+ Si-
7.61 7.93 7.97 8.19 7.93
1.57 1.28 1.19 1.02 1.26
0
2
4
6
8
10
X1 X2 X3 X4 X5
Separation
measures
Spillway
Si+ Si-