Top PDF Inconsistency and non-additive Choquet integration in the Analytic Hierarchy Process

Inconsistency and non-additive Choquet integration in the Analytic Hierarchy Process

Inconsistency and non-additive Choquet integration in the Analytic Hierarchy Process

In order to take into account some appropriate measure of inconsistency be- tween criteria which may be present in the main matrix A in modulating the weighted averaging scheme of the AHP, it is natural to extend the standard weighted mean aggregation to the more general framework of Choquet integra- tion. Comprehensive reviews of Choquet integration con be found in Grabisch and Labreuche [22, 23, 24], Grabisch, Kojadinovich, and Mayer [21], plus also Wang and Klir [49], Grabisch, Nguyen and Walker [27], Grabisch, Murofushi and Sugeno [26]. The Choquet integral is defined with respect to a non-additive capacity and corresponds to a large class of aggregation functions, including the classical weighted mean - the additive capacity case - and the ordered weighted means (OWA) - the symmetric capacity case. General reviews of aggregation functions can be found in Calvo, Mayor, and Mesiar [5], Beliakov, Pradera, and Calvo [2], Grabisch, Marichal, Mesiar, and Pap [25].
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Measuring the Sustainability of Cities in Turkey with the Analytic Hierarchy Process

Measuring the Sustainability of Cities in Turkey with the Analytic Hierarchy Process

DOI: 10.4236/jss.2019.74025 328 Open Journal of Social Sciences cator were sorted from the lowest to the highest. Then they were divided into three groups such as low, medium and high. After making the rating definitions for each indicator, the standard deviations of the series of the indicators were calculated. The alternatives, which are at the boundaries, have been moved to an upper or lower group according to the direction of the indicators whether the contribution of the indicator to sustainability is negative or positive. Since the weights of the rating definitions for each indicator are different, the experts have also compared the rating definitions. These weights were calculated by the geo- metric mean method and the inconsistency rates were also checked. All calcula- tions are below the critical value 0.10.
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Analytic hierarchy process-fuzzy sorting: an analytic hierarchy process based method for fuzzy classification in sorting problems

Analytic hierarchy process-fuzzy sorting: an analytic hierarchy process based method for fuzzy classification in sorting problems

21 provided in Section 3.1, and in particularly, in Phases 2 & 3. At this stage, the input of a decision-maker is necessary to decide upon these required parameters based on his/her expertise. Thus, we consulted with an expert having several years of professional experience and academic background on crime to obtain the required information. The expert originally filled in a questionnaire containing the key parameters characterizing each safety class and a pairwise comparison among the criteria to elicit their weights. The output is given as follows: Table 3 outlines the expert’s view on the profiles of each level of safety per criterion and Table 4 contains the pairwise comparisons of the criteria and the weights calculated with (1). Seemingly, ‘violent crimes’, followed by ‘weapons’ and ‘drug-related’ crimes are by far the most weighted crimes. This seems reasonable, being the most dangerous crimes for the public. The inconsistency ratio, computed as in (2), is 0.06, below the 0.1 threshold we had set, and thus we proceeded with the subsequent steps of the analysis.
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Analytic hierarchy process for new product development

Analytic hierarchy process for new product development

Once  again,  the  authors  strongly  believe  in  the  value  of  the  AHP  approach  to  be  used  in  the  processes  of  NPD.  Too often top managers select new project characteristics  under  uncertain  and  risky  conditions,  with  a  lack  of  detailed  information  [37].  Griffin  [38]  shows  that  about  40%  of  firms  do  not  use  formal  NPD  processes  and  Cooper  et  al.  [39]  demonstrated  that  almost  40%  of  investigated  firms  use  a  subjective  scoring  approach  in  NPD projects. Nonetheless, despite the great potential of  AHP  in  NPD,  some  technical  problems  must  be  addressed.  Low  consistency  may  arise  in  the  interviews  of  a  potential  customer,  expressing  opinions  on  future  products that may be much different from those actually  used. In the following paragraph, the problem observed  during  an  NPD  project  is  described.  The  consistent  questionnaires  resulted  to  be  so  few  that  any  data  analysis would have been undermined by a low statistical  significance. As a consequence, one should reflect if it is  preferable  to  process  only  a  few  consistent  interviews  –  following the most consolidated practice in AHP results  analysis, but loosing statistical significance – or consider  the entire population of respondents, which means taking  into  account  the  opinion  of  all  the  potential  market,  including  those  people  who  seem  to  be  less  confident  with the product analysed. In other words, should a non‐ consistent  potential  customer  be  excluded  from  the  analysis  for  his  inconsistency  or  should  he  be  included  because,  after  all,  he  is  still  a  potential  consumer?  This  question is prominent if related to the development of an  impulse  buying  good,  that  is  in  cases  where  customers  typically  decide  to  buy  products  without  a  rational  comparison  between  their  needs  and  the  product  features, instead just following their emotions [40].    
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On teaching the analytic hierarchy process

On teaching the analytic hierarchy process

Some students try to adjust their pairwise comparisons to achieve an IR =0. The instructor should discourage this practice. A little inconsistency is acceptable. It allows Aexibility in capturing the student’s understanding of the decision problem, as reAected by the pairwise comparison data. It is better to represent the user’s view, even if it is a little inconsistent, than to be consistent (IR=0) and wrong! One would not expect to 5nd a 7 × 7 pairwise comparison matrix to be perfectly consistent. The diBerent types of sensitivity analyses embedded in EC can be of great value in looking for inconsistencies, as well as answering “What if ?” questions. The instructor should demonstrate the sensitivity options as part of the choosing the best automobile exercise given as EX1 in the working paper. For example, by using the dynamic sensitivity option in EC, the student can readily observe how the solution changes as the weights on price are varied.
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Introduction to the Analytic Hierarchy Process

Introduction to the Analytic Hierarchy Process

Many other inconsistency indices have been proposed. For instance, Koczkodaj [80] proposed an inconsistency index for matrices of order three which was later extended to matrices of greater order [48]. Golden and Wang formulated an index which considers a metric between the normalized columns of the matrix and the priority vector obtained either with the eigenvector or the geometric mean method [63]. Cavallo and D’Apuzzo proposed an interpretation of pairwise comparison matrices using group theory and introduced their own index [35, 36]. Barzilai, first transformed the entries of the matrix by means of a logarithmic function and then proposed another index [9]. Gass and Rapcs´ak [62] defined an index based on the singular value decomposition of matrices. Furthermore, consider that even the objective functions of the optimization problems of the logarithmic least squares (2.3) and the least squares (2.4) used in § 2.1 to derive the priority vector can be considered inconsistency index. The interested reader can refer to a survey paper with numerical tests on various indices [26].
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Analysis of decisions made using the analytic hierarchy process

Analysis of decisions made using the analytic hierarchy process

The first issue commonly reported in literature on the AHP has to do with Dr. Saaty’s choice of a linear 1 to 9 scale for use in the pairwise comparisons. The main issue descends from the possible inconsistency generated from comparisons. For instance, if factor A is determined to be slightly more important than B (3), and B is determined to be slightly more important than C (3), then to assure complete consistency, A would need to be rated as absolutely more important than C (3x3 = 9). This is a big jump for two factors that are only slightly more important than each other. Following logically, if any two comparisons, A to B and B to C, are more than slightly more important, the resultant third comparison, A to C, will induce inconsistency, since 9 is the highest rating possible for any comparison. In Some Comments on the Analytic Hierarchy Process, R.D. Holder proposes the use of an exponential scale to resolve the issue (Holder, 1990, pg. 1073-1074). Essentially, one needs to determine the minimum discernible difference in factors, α. Then, comparisons become exponential factors of α:
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The Analytic Hierarchy Process: A Methodology for Win-Win Management

The Analytic Hierarchy Process: A Methodology for Win-Win Management

After forming the preference matrices, the process moves to the third step of deriving relative weights for the various elements. The relative weights of the elements of each level with respect to an element in the next higher level are computed as the components of the normalized eigenvector associated with the largest eigenvalue of their comparison matrix. The composite weights of the decision alternatives are then determined by aggregating the weights throughout the hierarchy. This is done by following a path from the top of the hierarchy to each alternative at the lowest level and multiplying the weights along each segment of the path. The outcome of this aggregation is a normalized vector of the overall weights of the options. The reader interested in the mathematical aspects of this procedure is referred to Saaty (1996 & 2004).
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Implementation of the Analytic Hierarchy Process for Student Profile  Analysis

Implementation of the Analytic Hierarchy Process for Student Profile Analysis

Decision making is a process that is always encountered in practical life. From the simple choice of a product to the adoption of one strategy or another, the question remains the same: how to make the right choice by taking into account the specifici- ties of the criteria that influence the decision-making process. Nowadays, multi- criteria decision making is present in almost all situations and domains, for example, in industry, business, medicine, etc.

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An Application of Analytic Hierarchy Process in Vehicle Routing Problem

An Application of Analytic Hierarchy Process in Vehicle Routing Problem

of locations on the road network. Associated with vertex i ∈ V\ {0} is a non-negative demand di. The parameter cij represents a non-negative cost (traveling cost in this case) between vertices i and j. The parameters K and Uk are the number of vehicles and the capacity of vehicle k, respectively. A three-index integer programming formulation is presented here where binary variables xijk denotes the number of times arc (i,j) ∈ A is traversed by vehicle k (k = 1,…,K) in the optimal solution. In addition, there are binary variables yik (i ∈ V; k = 1,…,K) that take a value of 1 if vertex i is visited by vehicle k in the optimal solution and take a value of 0, otherwise. The mathematical formulation of the problem is as follows
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The analytic hierarchy process in decisionmaking for caprine health programmes

The analytic hierarchy process in decisionmaking for caprine health programmes

After consulting experts from both the public and academic sectors, it was found that the impact of the diseases overrides the economic feasibility of carrying out control measures and t[r]

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The analytic hierarchy process applied to maintenance strategy selection

The analytic hierarchy process applied to maintenance strategy selection

In designing the AHP hierarchical tree, the aim is to develop a general framework that satisfies the needs of the analysts to solve the selection problem of the best mainte- nance policy. The AHP starts by breaking down a complex, multi-criteria problem into a hierarchy where each level comprises a few manageable elements which are then broken down into another set of elements. Considering the critical aspect of this step for the AHP methodology, the structure has been created following suggestions from the maintenance and production staff of the oil refinery (Fig. 1). The AHP hierarchy developed in this study is a five-level tree in which the top level represents the main objective of maintenance selection and the lowest level comprises the alternative maintenance policies.
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Some Notes on Inconsistence and Indecisiveness in the Analytic Hierarchy Process

Some Notes on Inconsistence and Indecisiveness in the Analytic Hierarchy Process

Abstract: The aim of the article is to introduce a mathematical concept of indecisiveness into the analytic hierarchy process (AHP) framework. Indecisiveness can be useful in two ways: first, decision makers with high indecisiveness (higher than a given threshold value) can be excluded from a decision making process in its early stages as low-competent and replaced by other, more competent DMs; second, indecisiveness along with consistency index C.I. can be used for the calculation of (aposteriori) DMs’ weights without any additional information about DMs’ age, formal knowledge, social status, etc. The proposed approach is demonstrated on examples.
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The analytic hierarchy process for the decision tree with multiple criteria

The analytic hierarchy process for the decision tree with multiple criteria

When the decision problem is homogenous, described by decision table, its hierarchy is complete. However, the problem hierarchy (not complete) can also be construct- ed for the non-homogeneous problems when some alter- natives are not influenced by all states of nature, when some combinations alternative/state of nature are unrea- sonable and when more decision modes exist.

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Using the analytic hierarchy process in evaluating decision alternatives

Using the analytic hierarchy process in evaluating decision alternatives

In the broadest terms, we can isolate two types of hierarchy: structural and func- tional [12, p. 30–31]. A structural hierarchy indicates the relationships between the component parts of complex systems, where these relationships are understood as an arrangement in terms of structural properties (e.g. shape, volume, colour, age). Struc- tural hierarchy mirrors the way in which the human mind analyses complexity by dis- tinguishing kinds, classes, groups, sub-groups, types, sets, and subsets. Functional hierarchy, on the other hand, appears in the relationships between elements on a given hierarchical level. Elements on a given level are thus not completely equivalent. Functional hierarchy means that there are also relationships between them of the type: “less important”, “more important”.
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Measuring Educational Service Quality Using Analytic Hierarchy Process

Measuring Educational Service Quality Using Analytic Hierarchy Process

The main objective of this study was to examine the process of parental evaluation of educational service quality within public secondary schools by identifying, prioritizing and examining the criteria used by parents. The AHP methodology was used in the study as it draws on both qualitative and quantitative information. The general results of the study demonstrate that parents indicate the individual development as the most important quality factor in public secondary schools. Educational development takes the second place with a slight difference. Parents, then, seek more specific items under physical maintenance criterion. The findings show that despite the serious lack of financial resources, laboratory maintenance, computer and library maintenances in Turkish public schools (Gedikoğlu, 2005), parents did not evaluate physical maintenance as a prior factor of service quality. The significant point that must be underlined at this point is that parents treat individual development as important as and even more important than educational development. This refers to the responsibility of educational institutions, raising students with strong social, cultural, sporting, communication, free and creative thinking skills. Developing just academic skills of students is not enough. Students must also be inspired to develop their intellectual skills. In Turkey, up to now, transmitting academic skills and enhancing quantitative abilities have been considered to be important. Developing social skills of students has generally been neglected (Özden, 1999 ). However, the current study demonstrates that the parents’ view has been changing. In other words, parents demand an education program providing individual and educational skills simultaneously. This might mean that there is an increase in social awareness about raising fully equipped individuals for the future of the nation. As Guile (2001) noted in the information or knowledge–based economies of the future, people will become capital rather than traditional factors of production. Therefore, full intellectual development of each student is a must. Educational institutions have the most important duty in this respect. However, the results of the study are more important and carefully taken into consideration by policy makers as they are decision makers of service quality and its improvement in education sector.
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Integration of fuzzy reasoning approach (FRA) and fuzzy analytic hierarchy process (FAHP) for risk assessment in mining industry

Integration of fuzzy reasoning approach (FRA) and fuzzy analytic hierarchy process (FAHP) for risk assessment in mining industry

For the application of FRA approach Fuzzy Logic tool box of MATLAB is used. To build the Fuzzy Inference System in MATLAB for the case under study, fuzzy inputs were decided. Thereafter fuzzy operator is designated for antecedents of given rule if there is more then one part in antecedent in any rule. Two operators either AND or OR can be used for this purpose. Then implication method is applied before which the rule weight is decided. The weight of rule is between 0 to 1, this weight is decided based upon the number given by the antecedent. The input for implication is the number provided by the antecedent and output of implication is a fuzzy set. After this output is aggregated. Aggregation is the process by which the fuzzy sets that represents the out puts of each rule are combined into a single fuzzy set. Then this fuzzy set is defuzzified and crisp output is obtained.
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DETERMINATION OF THE THERMAL HOTEL LOCATION: APPLICATION OF ANALYTIC HIERARCHY PROCESS

DETERMINATION OF THE THERMAL HOTEL LOCATION: APPLICATION OF ANALYTIC HIERARCHY PROCESS

AHP as a subjective and qualitative evaluation method has been broadly used for environmental planning, evaluation of transportation alternatives, evaluation of factors inf[r]

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The Determination of University Selection Based Upon Analytic Hierarchy Process

The Determination of University Selection Based Upon Analytic Hierarchy Process

Abstract: The most important factor in career planning of a person is to direct him depending upon his features. The best way of choosing career is to compare the wishes of a person with the requirements of that career so that he can decide the best one. Particularly, those who think to have a university education for their careers come across difficulties while deciding on their career path on account of the fact that the global world can offer various opportunities for education in a great many places. The student must choose by taking into account some criteria. As an example, several factors play a crucial role in this process such as the academic success of the university, the working opportunities provided, the distance of the university to the hometown of the student, the economic status of that city, the facilities of accommodation. Considering all these factors, the student should give an optimal decision. In this context, the common decision including both the personal different opinions and convincing for all is strongly needed. AHP (Analytic Hierarchy Process) has gained a very big momentum at these kind of situations.
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Pharmaceutical supply chain risk assessment in Iran using analytic hierarchy process (AHP) and simple additive weighting (SAW) methods

Pharmaceutical supply chain risk assessment in Iran using analytic hierarchy process (AHP) and simple additive weighting (SAW) methods

Typically, hierarchy is structured from the top (goal) to the end (criteria or alternative) which the goal is finding the most appropriate criteria or alternative among others based on the weight of each criteria or alternative. In fact the group AHP method based on the pairwise comparison tries to find differences be- tween criteria or alternatives and obtain the weight of each objective. The goal of this research was to evalu- ate and prioritize management of supply chain func- tions from a pharmaceutical production company perspective. Below steps are defined based on the group AHP approach, to prioritize pharmaceutical supply chain functions:
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