This document discusses measuring the importance and weight of decision makers in group decision making processes. It introduces a method to calculate weights for decision makers based on the number of iterations their pairwise comparison matrices take to reach convergence when using the eigenvector method for criteria weighting. A case study applies this method to determine the weights of three decision makers (A, B, and C) who provide pairwise comparison matrices for weighting four criteria related to selecting a form. The number of iterations for each decision maker's matrix to converge determines their absolute and relative weights in the group decision making process.
Multi criteria decision making in spatial data analysisPreeti Tiwari
There are a number of multi-criteria methods
that can be utilized to facilitate individual or
group decision-making:
1. Analytic Hierarchy Process (AHP)
2. AHP Combined Method
3. Fuzzy AHP
4. Fuzzy AHP Combined
5. Fuzzy AHP Group
6. Group Evaluation Method
7. Weighted Sum Method (WSM)
8. Weighted Product Method (WPM)
Classification accuracy analyses using Shannon’s EntropyIJERA Editor
There are many methods for determining the Classification Accuracy. In this paper significance of Entropy of
training signatures in Classification has been shown. Entropy of training signatures of the raw digital image
represents the heterogeneity of the brightness values of the pixels in different bands. This implies that an image
comprising a homogeneous lu/lc category will be associated with nearly the same reflectance values that would
result in the occurrence of a very low entropy value. On the other hand an image characterized by the
occurrence of diverse lu/lc categories will consist of largely differing reflectance values due to which the
entropy of such image would be relatively high. This concept leads to analyses of classification accuracy.
Although Entropy has been used many times in RS and GIS but its use in determination of classification
accuracy is new approach.
Multi criteria decision making in spatial data analysisPreeti Tiwari
There are a number of multi-criteria methods
that can be utilized to facilitate individual or
group decision-making:
1. Analytic Hierarchy Process (AHP)
2. AHP Combined Method
3. Fuzzy AHP
4. Fuzzy AHP Combined
5. Fuzzy AHP Group
6. Group Evaluation Method
7. Weighted Sum Method (WSM)
8. Weighted Product Method (WPM)
Classification accuracy analyses using Shannon’s EntropyIJERA Editor
There are many methods for determining the Classification Accuracy. In this paper significance of Entropy of
training signatures in Classification has been shown. Entropy of training signatures of the raw digital image
represents the heterogeneity of the brightness values of the pixels in different bands. This implies that an image
comprising a homogeneous lu/lc category will be associated with nearly the same reflectance values that would
result in the occurrence of a very low entropy value. On the other hand an image characterized by the
occurrence of diverse lu/lc categories will consist of largely differing reflectance values due to which the
entropy of such image would be relatively high. This concept leads to analyses of classification accuracy.
Although Entropy has been used many times in RS and GIS but its use in determination of classification
accuracy is new approach.
The selection of the best employees is one of the process of evaluating how well the performance of the employees is adjusted to the standards set by the company and usually done by top management such as General Manager or Director. In general, the selection of the best employees is still perform manually with many criteria and alternatives, and this usually make it difficult top managerial making decisions as well as the selection of the best employees periodically into a long and complicated process. Therefore, it is necessary to build a decision support system that can help facilitate the decision maker in determining the best choice based on standard criteria, faster, and more objective. In this research, the computational method of decision-making system used is Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS). The criteria used in the selection of the best employees are: job responsibilities, work discipline, work quality, and behaviour. The final result of the global priority value of the best employee candidates is used as the best employee selection decision making tool by top management.
Stages of decision making done by the manager is a crucial stage. Given the resulting decisions affect the sustainability of the organization, then many managers use systems that can support the resulting decisions. This system is known as the decision support system, which applies to solving a problem, using methods such as ELECTRE, Promethee, SAW, TOPSIS. Using decision support systems makes it easy for decision makers to add new data, change data and make decisions more efficiently. In this article, the method used is Technique for Order Preference by Similarity to Ideal Solution (TOPSIS).
Decision support system is an interactive system to support decision-making process through the alternatives derived from
the processing of data, information and design models. In this research will build a decision support system modeling for the
determination of admission scholarship, as long as this problem of determining admission scholarship often become obstacles in
distribution and is not directed at the destination as expected. Therefore, in order to give a better result and overcome obstacles in the
distribution of scholarships. The problems of determining admission scholarship will be resolved through Fuzzy approach to the Analytic Hierarchy Process (AHP) is modeled in a decision support system modeling. Where Fuzzy will perform the functions of representation based membership in the assessment criteria. So the results given Fuzzy will be approached with the weight vector
given by the Analytic Hierarchy Process (AHP) which would then be carried out by the ranking process Analiytic Hierarchy Process (AHP) to determine the best alternative will be selected as scholarship recipients. After Fuzzy AHP approach in modeling decision support systems, particularly in the determination of admission scholarships and given very good results and focus on the goal as expected.
A NOVEL APPROACH FOR FEATURE EXTRACTION AND SELECTION ON MRI IMAGES FOR BRAIN...cscpconf
Feature extraction is a method of capturing visual content of an image. The feature extraction is
the process to represent raw image in its reduced form to facilitate decision making such as
pattern classification. The objective of this paper is to present a novel method of feature
selection and extraction. This approach combines the Intensity, Texture, shape based features
and classifies the tumor as white matter, Gray matter, CSF, abnormal and normal area. The
experiment is performed on 140 tumor contained brain MR images from the Internet Brain
Segmentation Repository. PCA and Linear Discriminant Analysis (LDA) were applied on the
training sets. The Support Vector Machine (SVM) classifier served as a comparison of
nonlinear techniques Vs linear ones. PCA and LDA methods are used to reduce the number of
features used. The feature selection using the proposed technique is more beneficial as it
analyses the data according to grouping class variable and gives reduced feature set with high classification accuracy.
Application of Fuzzy Analytic Hierarchy Process and TOPSIS Methods for Destin...ijtsrd
Destination selection is one of the most become an extremely popular. Sometimes the terms tourism and tourism are used pejoratively to indicate a shallow interest in the societies or islands that traveler's tour. This system presents the use of fuzzy AHP and TOPSIS for deciding on the selection of destination as like the selection of island. In this system, eight countries that include in South East Asia Thailand, Singapore, Malaysia, Indonesia, Philippine, Vietnam, Cambodia, Brunei are used. At first, the user can choose the specific country to decide the island of these countries and their preferences attraction, environment, accommodation, transportation, restaurant, activity, entertainment and other facilities are taken as inputs and then display the list of alternatives that matched with user's preferences. Fuzzy analytic hierarchy process is used in determining the weight of criteria and alternatives. Technique for Order Preference by Similarity to Ideal Solution TOPSIS method is used for determining the final ranking of the alternatives. Finally, this system shows the list of destinations depend on user's preferences. Hnin Min Oo | Su Hlaing Hnin "Application of Fuzzy Analytic Hierarchy Process and TOPSIS Methods for Destination Selection" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd27975.pdfPaper URL: https://www.ijtsrd.com/computer-science/data-processing/27975/application-of-fuzzy-analytic-hierarchy-process-and-topsis-methods-for-destination-selection/hnin-min-oo
Data Mining is an analytic process designed to explore data in search of consistent patterns and systematic relationships between variables, and then to validate the results by applying the patterns found to a new subset of data. Data mining is often described as the process of discovering patterns, correlations, trends or relationships by searching through a large amount of data stored in repositories, databases, and data warehouses. Diabetes, often referred to by doctors as diabetes mellitus, describes a group of metabolic diseases in which the person has high blood [3] glucose (blood sugar), either because insulin production is insufficient, or because the body's cells do not respond properly to insulin, or both. This project helps in identifying whether a person has diabetes or not, if predicted diabetic[4] the project suggest measures for maintaining normal health and if not diabetic it predicts the risk of getting diabetic. In this project Classification algorithm was used to classify the Pima Indian diabetes dataset. Results have been obtained using Android Application.
The amount of information in the form of features and variables available to machine learning algorithms is ever increasing. This can lead to classifiers that are prone to overfitting in high dimensions, high dimensional models do not lend themselves to interpretable results, and the CPU and memory resources necessary to run on high-dimensional datasets severly limit the applications of the approaches.
Variable and feature selection aim to remedy this by finding a subset of features that in some way captures the information provided best.
In this paper we present the general methodology and highlight some specific approaches.
Estimation of Reliability in Multicomponent Stress-Strength Model Following B...IJERA Editor
The stress-strength model for system reliability for multicomponent system when a device under consideration is a combination of 𝒦 usually independent components with strengths 𝑋1,𝑋2,…,𝑋𝒦 and each component experiencing a common stress 𝑌. The system is regarded as alive if at least𝒮 out of 𝒦 𝒮<𝒦 strengths exceed the stress. The reliability of such system is obtained when the stress and strengths are assumed to have Burr XII distributions with common shape parameter(𝑐). The research methodology adapted here is to estimate the parameters by using maximum likelihood method. Based on different types of ranked set sampling, the reliability estimators are obtained when samples drawn from strength and stress distributions. Efficiencies of reliability estimators based on ranked set sampling, median ranked set sampling, extreme ranked set sampling and percentile ranked set sampling are calculated with respect to reliability estimators based on simple random sampling procedure.
Multi-Population Methods with Adaptive Mutation for Multi-Modal Optimization ...ijscai
This paper presents an efficient scheme to locate multiple peaks on multi-modal optimization problems by
using genetic algorithms (GAs). The premature convergence problem shows due to the loss of diversity,
the multi-population technique can be applied to maintain the diversity in the population and the
convergence capacity of GAs. The proposed scheme is the combination of multi-population with adaptive
mutation operator, which determines two different mutation probabilities for different sites of the
solutions. The probabilities are updated by the fitness and distribution of solutions in the search space
during the evolution process. The experimental results demonstrate the performance of the proposed
algorithm based on a set of benchmark problems in comparison with relevant algorithms.
The selection of the best employees is one of the process of evaluating how well the performance of the employees is adjusted to the standards set by the company and usually done by top management such as General Manager or Director. In general, the selection of the best employees is still perform manually with many criteria and alternatives, and this usually make it difficult top managerial making decisions as well as the selection of the best employees periodically into a long and complicated process. Therefore, it is necessary to build a decision support system that can help facilitate the decision maker in determining the best choice based on standard criteria, faster, and more objective. In this research, the computational method of decision-making system used is Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS). The criteria used in the selection of the best employees are: job responsibilities, work discipline, work quality, and behaviour. The final result of the global priority value of the best employee candidates is used as the best employee selection decision making tool by top management.
Stages of decision making done by the manager is a crucial stage. Given the resulting decisions affect the sustainability of the organization, then many managers use systems that can support the resulting decisions. This system is known as the decision support system, which applies to solving a problem, using methods such as ELECTRE, Promethee, SAW, TOPSIS. Using decision support systems makes it easy for decision makers to add new data, change data and make decisions more efficiently. In this article, the method used is Technique for Order Preference by Similarity to Ideal Solution (TOPSIS).
Decision support system is an interactive system to support decision-making process through the alternatives derived from
the processing of data, information and design models. In this research will build a decision support system modeling for the
determination of admission scholarship, as long as this problem of determining admission scholarship often become obstacles in
distribution and is not directed at the destination as expected. Therefore, in order to give a better result and overcome obstacles in the
distribution of scholarships. The problems of determining admission scholarship will be resolved through Fuzzy approach to the Analytic Hierarchy Process (AHP) is modeled in a decision support system modeling. Where Fuzzy will perform the functions of representation based membership in the assessment criteria. So the results given Fuzzy will be approached with the weight vector
given by the Analytic Hierarchy Process (AHP) which would then be carried out by the ranking process Analiytic Hierarchy Process (AHP) to determine the best alternative will be selected as scholarship recipients. After Fuzzy AHP approach in modeling decision support systems, particularly in the determination of admission scholarships and given very good results and focus on the goal as expected.
A NOVEL APPROACH FOR FEATURE EXTRACTION AND SELECTION ON MRI IMAGES FOR BRAIN...cscpconf
Feature extraction is a method of capturing visual content of an image. The feature extraction is
the process to represent raw image in its reduced form to facilitate decision making such as
pattern classification. The objective of this paper is to present a novel method of feature
selection and extraction. This approach combines the Intensity, Texture, shape based features
and classifies the tumor as white matter, Gray matter, CSF, abnormal and normal area. The
experiment is performed on 140 tumor contained brain MR images from the Internet Brain
Segmentation Repository. PCA and Linear Discriminant Analysis (LDA) were applied on the
training sets. The Support Vector Machine (SVM) classifier served as a comparison of
nonlinear techniques Vs linear ones. PCA and LDA methods are used to reduce the number of
features used. The feature selection using the proposed technique is more beneficial as it
analyses the data according to grouping class variable and gives reduced feature set with high classification accuracy.
Application of Fuzzy Analytic Hierarchy Process and TOPSIS Methods for Destin...ijtsrd
Destination selection is one of the most become an extremely popular. Sometimes the terms tourism and tourism are used pejoratively to indicate a shallow interest in the societies or islands that traveler's tour. This system presents the use of fuzzy AHP and TOPSIS for deciding on the selection of destination as like the selection of island. In this system, eight countries that include in South East Asia Thailand, Singapore, Malaysia, Indonesia, Philippine, Vietnam, Cambodia, Brunei are used. At first, the user can choose the specific country to decide the island of these countries and their preferences attraction, environment, accommodation, transportation, restaurant, activity, entertainment and other facilities are taken as inputs and then display the list of alternatives that matched with user's preferences. Fuzzy analytic hierarchy process is used in determining the weight of criteria and alternatives. Technique for Order Preference by Similarity to Ideal Solution TOPSIS method is used for determining the final ranking of the alternatives. Finally, this system shows the list of destinations depend on user's preferences. Hnin Min Oo | Su Hlaing Hnin "Application of Fuzzy Analytic Hierarchy Process and TOPSIS Methods for Destination Selection" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd27975.pdfPaper URL: https://www.ijtsrd.com/computer-science/data-processing/27975/application-of-fuzzy-analytic-hierarchy-process-and-topsis-methods-for-destination-selection/hnin-min-oo
Data Mining is an analytic process designed to explore data in search of consistent patterns and systematic relationships between variables, and then to validate the results by applying the patterns found to a new subset of data. Data mining is often described as the process of discovering patterns, correlations, trends or relationships by searching through a large amount of data stored in repositories, databases, and data warehouses. Diabetes, often referred to by doctors as diabetes mellitus, describes a group of metabolic diseases in which the person has high blood [3] glucose (blood sugar), either because insulin production is insufficient, or because the body's cells do not respond properly to insulin, or both. This project helps in identifying whether a person has diabetes or not, if predicted diabetic[4] the project suggest measures for maintaining normal health and if not diabetic it predicts the risk of getting diabetic. In this project Classification algorithm was used to classify the Pima Indian diabetes dataset. Results have been obtained using Android Application.
The amount of information in the form of features and variables available to machine learning algorithms is ever increasing. This can lead to classifiers that are prone to overfitting in high dimensions, high dimensional models do not lend themselves to interpretable results, and the CPU and memory resources necessary to run on high-dimensional datasets severly limit the applications of the approaches.
Variable and feature selection aim to remedy this by finding a subset of features that in some way captures the information provided best.
In this paper we present the general methodology and highlight some specific approaches.
Estimation of Reliability in Multicomponent Stress-Strength Model Following B...IJERA Editor
The stress-strength model for system reliability for multicomponent system when a device under consideration is a combination of 𝒦 usually independent components with strengths 𝑋1,𝑋2,…,𝑋𝒦 and each component experiencing a common stress 𝑌. The system is regarded as alive if at least𝒮 out of 𝒦 𝒮<𝒦 strengths exceed the stress. The reliability of such system is obtained when the stress and strengths are assumed to have Burr XII distributions with common shape parameter(𝑐). The research methodology adapted here is to estimate the parameters by using maximum likelihood method. Based on different types of ranked set sampling, the reliability estimators are obtained when samples drawn from strength and stress distributions. Efficiencies of reliability estimators based on ranked set sampling, median ranked set sampling, extreme ranked set sampling and percentile ranked set sampling are calculated with respect to reliability estimators based on simple random sampling procedure.
Multi-Population Methods with Adaptive Mutation for Multi-Modal Optimization ...ijscai
This paper presents an efficient scheme to locate multiple peaks on multi-modal optimization problems by
using genetic algorithms (GAs). The premature convergence problem shows due to the loss of diversity,
the multi-population technique can be applied to maintain the diversity in the population and the
convergence capacity of GAs. The proposed scheme is the combination of multi-population with adaptive
mutation operator, which determines two different mutation probabilities for different sites of the
solutions. The probabilities are updated by the fitness and distribution of solutions in the search space
during the evolution process. The experimental results demonstrate the performance of the proposed
algorithm based on a set of benchmark problems in comparison with relevant algorithms.
Grant Selection Process Using Simple Additive Weighting Approachijtsrd
Selection of educational grant is a key success factor for student learning and academic performance. Among popular methods, this paper contributes a real problem of selecting educational grant using data of grant application forms by one of the multi criteria decision making model, SAW method. This paper introduces nine criteria that are qualitative and positive for selecting grant for the students amongst fifteen application forms and also ranking them. Finally, the proposed method is demonstrated in a case study on selecting educational grant for students. Kyi Kyi Myint | Tin Tin Soe | Myint Myint Toe ""Grant Selection Process Using Simple Additive Weighting Approach"" 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/ijtsrd25169.pdf
Paper URL: https://www.ijtsrd.com/computer-science/simulation/25169/grant-selection-process-using-simple-additive-weighting-approach/kyi-kyi-myint
Analytical Hierarchical Process has been used as a useful methodology for multi-criteria decision making environments with substantial applications in recent years. But the weakness of the traditional AHP method lies in the use of subjective judgement based assessment and standardized scale for pairwise comparison matrix creation. The paper proposes a Condorcet Voting Theory based AHP method to solve multi criteria decision making problems where Analytical Hierarchy Process (AHP) is combined with Condorcet theory based preferential voting technique followed by a quantitative ratio method for framing the comparison matrix instead of the standard importance scale in traditional AHP approach. The consistency ratio (CR) is calculated for both the approaches to determine and compare the consistency of both the methods. The results reveal Condorcet- AHP method to be superior generating lower consistency ratio and more accurate ranking of the criterion for solving MCDM problems.
This is a series of Capacity Building documents that was prepared by the Sudanese Youth Leadership Development Program.
هذه مجموعة من المقالات في مجالات تدريبية متعددة مناسبة للجمعيات الطوعية تم تطويرها بين عامي 2003-2008 للبرنامج السوداني لإعداد القيادات الشبابية
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.
Evaluation metric plays a critical role in achieving the optimal classifier during the classification training.
Thus, a selection of suitable evaluation metric is an important key for discriminating and obtaining the
optimal classifier. This paper systematically reviewed the related evaluation metrics that are specifically
designed as a discriminator for optimizing generative classifier. Generally, many generative classifiers
employ accuracy as a measure to discriminate the optimal solution during the classification training.
However, the accuracy has several weaknesses which are less distinctiveness, less discriminability, less
informativeness and bias to majority class data. This paper also briefly discusses other metrics that are
specifically designed for discriminating the optimal solution. The shortcomings of these alternative metrics
are also discussed. Finally, this paper suggests five important aspects that must be taken into consideration
in constructing a new discriminator metric.
AHP technique a way to show preferences amongst alternativesijsrd.com
This article presents a review of the applications of Analytic Hierarchy Process (AHP). AHP is a multiple criteria decision-making tool that has been used in almost all the applications related with decision-making. Decisions involve many intangibles that need to be traded off. The Analytic Hierarchy Process (AHP) is a theory of measurement through pairwise comparisons and relies on the judgements of experts to derive priority scales. It is these scales that measure intangibles in relative terms. The comparisons are made using a scale of absolute judgements that represents how much more; one element dominates another with respect to a given attribute. The judgements may be inconsistent, and how to measure inconsistency and improve the judgements, when possible to obtain better consistency is a concern of the AHP. The derived priority scales are synthesised by multiplying them by the priority of their parent nodes and adding for all such nodes. An illustration is also included.
PRIORITIZING THE BANKING SERVICE QUALITY OF DIFFERENT BRANCHES USING FACTOR A...ijmvsc
In recent years, India’s service industry is developing rapidly. The objective of the study is to explore the
dimensions of customer perceived service quality in the context of the Indian banking industry. In order to
categorize the customer needs into quality dimensions, Factor analysis (FA) has been carried out on
customer responses obtained through questionnaire survey. Analytic Hierarchy Process (AHP) is employed
to determine the weights of the banking service quality dimensions. The priority structure of the quality
dimensions provides an idea for the Banking management to allocate the resources in an effective manner
to achieve more customer satisfaction. Technique for Order Preference Similarity to Ideal Solution
(TOPSIS) is used to obtain final ranking of different branches.
Invited talk at the Focus Fortnight 8: ""The analysis of discrete choice experiments", organized by the Centre for Bayesian Statistics in Health Economics, University of Sheffield (UK), September, 2007.
In the problem of determining the proper level of sequence in a sorting problem, the Simple Additive Weighting (SAW) method is an easy-to-use technique. It can analyze cases based on the criteria used. The use of criteria values in this approach has an unlimited amount. The more criteria used, the higher the accuracy of the results obtained. There are two types of criteria, cost, and benefits. Cost is used if the higher the criterion value, the lower the chance to get the top score while the benefit is used if the higher the criterion value, the greater the chance to get the top position. This study explains how to apply SAW algorithm in solving sequence problems in various cases encountered. Using this method will solve decisions that can not be completed manually. It helps the admin in choosing the best decision in any particular instance.
2. Measuring the Importance and the Weight of Decision Makers in the Criteria
Weighting Activities of Group Decision Making Process 7
making, is a vital issue in most areas of scientific activities that includes, best alternative finding from
the set of available options (Asgharpour, 2006). Criteria weights change in decision making process,
those have important affect on results of the decision making (Asgharpour, 2006). Methods such as,
LINMAP, SMART, Eigenvector and something like that, use for finding the criteria weights.
In This paper, the eigenvector method use for criteria weighting and then the process of finding
weights use for decision makers importunacy in group decision making. Often seen these decision
makers in all methods of group decision making (even in voting methods) have same weight of
importance for participate in the decision making process that has its drawbacks.
Generally multiple criteria decision making, including several in facts as follow:
1. Criteria Recognition and evaluation.
2. Allocate weight to each criterion
3. Best alternative selection or ranking with one of MADM methods
4. Make sensitivity analysis operations.
2. Eigenvector Method for Criteria Weighting
There are some methods for criteria weighting in decision making process (toloie, 2006). Eigenvector
method is one of these methods that use in circumstancing which decision making matrix is not
available. This method is based on pair wise comparison (Saati, 1980). In pair wises comparison
method, criteria preference find by using below table (Table 1). The preference measurement scales are
shown on table 2.
Table 1: Criteria pair wise comparison matrix
Criteria Criterion 1 Criterion 2 Criterion j
Criterion 1 Criterion 1 to criterion 1 Criterion 1 to criterion 2 Criterion 1 to criterion j
Criterion 2 Criterion 2 to criterion 1 Criterion 2 to criterion 2 Criterion 2 to criterion j
Criterion i Criterion i to criterion 1 Criterion i to criterion 2 Criterion i to criterion j
Table 2: Pair wise comparison scales (i to j)
1 Equal preference
3 Poor preference
5 Strong preference
7 Very strong preference
9 Absolute preference
2,4,6,8 Intermediate preference
Matrix that introduced in Table 1, always, is a square matrix and criteria that shown in rows
and column will be the same. As is clear, main diameter values of the matrix will equal one, because in
fact, it shows relative value and importance of each criterion to own. What happened for the rest matrix
members? Let's suppose that criterion1 has strong preference to criterion 2 then decision maker should
be settling 5, in cellule 12 that is calling f12. However, filling the matrix should be noticing two
important following principal:
• Reciprocal principal: If suppose that the value preference of ith criterion, to jth criterion is a
(means decision maker preference ith criterion to jth criterion, a times), logically, decision maker
have to prefer 1 / a, jth criterion to ith criterion.
1
ij 1 2 3
= i, j = , , ,....,n
(1)
f
f
ji
• Consistency principal: decision maker should be fully remembering that if:
Criterion 1 Criterion 2
3. 8 Abbas Toloie-Eshlaghy and Ebrahim Nazari Farokhi
And
Criterion 2 Criterion 3
Then:
Criterion 1 Criterion 3
In total consistency, have to:
f f f i, j ,k , , ,....,n ik kj ij = = 1 2 3 (2)
In addition, the decision maker should be sure that if:
Preference of 1th criterion to 2th criterion is equal 3 and also, preference of 2th criterion to 3th
criterion is equal 2, then the preference of 1th criterion to 3th criterion have to be 2*3 = 6.
The second principal, in fact, formed the basic and core concepts of this article. After establish
pair wise matrix, by using following formula, the matrix must be iterated multiple times, to finally be
close to convergence vector.
k
D .e
e .D .e
W Lim
t k
j
= (3)
k ®¥
k Î Integer
That:
Wj is jth weights vector
D is initial pair wise comparison matrix
e unit column vector that all elements are equal 1
et is transposing matrix of e
Number of iterations depends on the following two cases:
• If the number of criteria increases then the number of iterations of matrix for achieving to
convergence vector also increases (However, this relation is not linear).
• If the decision maker inconsistency increase then the number of matrix iterations also
increase.
In actual conditions, decision makers have different levels of accessible information, thinking
capabilities and experience. It is impossible that in decision making process, two individual decision
makers have same judgment. However, it happened by different reasons, subject to the talents and
capabilities of different people, cannot achieve to the same access of resources of information and so
on (Asgharpour, 2006). Therefore, the decision makers’ pair wise comparison matrixes, always, are
inconsistence. It seems that if the decision makers be inconsistence in decision making process, then
number of iterations to reach a convergence vector increase. So, the number of iterations maybe a good
basis, for measuring accuracy and consistency of decision makers. Calculated weights and importance
can be used in group decision making process method such as, BORDA technique, DEMATEL
technique, and something like that.
When the number of iterations, for each person by using the eigenvector method, achieved and
since the sum of weights of participants in the group decision making process, should be equal to 1
(because the relative importance of decision makers should consider) , then by using the following
relation, weight and importance of each decision maker could be calculate:
Absolute weight of each decision maker = 1 – (number of iterations for each decision makers
/total number of iterations for all decision makers) (4)
And, finally by using probability scale less method:
Relative weight of each decision maker = absolute weight of each decision maker / sum of all
decision maker absolute weights (5)
3. Case Study
In same conditions of space, location and time (for controlling the circumstances), following decision
matrix is completed by decision makers. Since this section of paper, takes a case study to identify the
4. Measuring the Importance and the Weight of Decision Makers in the Criteria
Weighting Activities of Group Decision Making Process 9
level of matrix inconsistency, so, the type of criteria are not important. Also for achieving more
effective visual perception, some forms considered as criteria. Therefore decision making process for
criteria weighting followed with 4 criteria as below:
And then, pair wise comparison matrix must be as follows:
Table 3: Pair wise comparison matrix for form selection
criteria
In this case study, three decision makers, play his role for measuring weights of them. The three
people have shown with capital letters A, B and C. The completed matrix for each decision maker,
shown as follows:
Table 4: Completed pair wise comparison matrix for decision maker A
criteria
1 5 7 4
1/5 1 1/3 8
1/7 3 1 2
1/4 1/8 1/2 1
Table 5: Completed pair wise comparison matrix for decision maker B
criteria
1 6 9 4
1/6 1 3 7
1/9 1/3 1 8
1/4 1/7 1/8 1
5. 10 Abbas Toloie-Eshlaghy and Ebrahim Nazari Farokhi
Table 6: Completed pair wise comparison matrix for decision maker C
criteria
1 2 3 4
1/2 1 5 6
1/3 1/5 1 4
1/4 1/6 1/4 1
After obtaining the pair wise comparison matrix for each decision maker, regards to formula
(2), iterations must be calculated to achieving convergence vector. In this article MATLAB software
used for to this purpose. For decision maker A, for example:
Input data:
D = [1 5 7 4; 1/5 1 1/3 8; 1/7 3 1 2; 1/4 1/8 1/2 1];
e = [1; 1; 1; 1];
et =[1 1 1 1];
Then, in:
First iteration:
W1= (D^1*e)/ ( et *D^1*e)
W1 =
0.4920
0.2759
0.1778
0.0543
And finally after 8 iterations:
W8= (W^8*e)/ (et *W^8*e)
W8 =
0.5860
0.1708
0.1792
0.0639
Weights that obtained in the seventh iteration, identically repeated in eighth iteration. Therefore
for decision maker A, number of iterations to achieve convergence vector is equal 8.
Just like the above steps, for decision maker B, number of iterations is equal 7, and for decision
maker C, number of iterations is equal 5.
Now according formula (4) and (5) will be:
Weight of decision maker A:
.
1 .
0 3
0 6
. =
0 6 0 65 0 75
0 6
8
20
. + . +
.
= − =
Weight of decision maker B:
.
1 .
0 325
0 65
. =
0 6 0 65 0 75
0 65
7
20
. + . +
.
= − =
Weight of decision maker C:
.
1 .
0 375
0 75
. =
0 6 0 65 0 75
0 75
5
20
. + . +
.
= − =
Therefore, decision maker C has the highest weight and decision maker A has with the lowest
weight and so, these weights show decision maker importance in decision making group.
6. Measuring the Importance and the Weight of Decision Makers in the Criteria
Weighting Activities of Group Decision Making Process 11
4. Conclusion
Since the criteria weighting in decision making process, often, have done by humans and decision
makers, so, inconsistent decision makers have less weight and also the consistent decision makers have
more weight. In this article, by using eigenvector weighting method based on pair wise comparison, a
new method introduced to demonstrate the importance of decision makers.
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