scholarly journals Characterization of Consistent Completion of Reciprocal Comparison Matrices

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Julio Benítez ◽  
Laura Carrión ◽  
Joaquín Izquierdo ◽  
Rafael Pérez-García
Keyword(s):  
Pairwise Comparison ◽  
Explicit Calculation ◽  
Frobenius Norm ◽  
Analytic Hierarchy ◽  
Linear System Of Equations ◽  
Decision Aiding ◽  
The Matrix ◽  
Decision Making Processes ◽  
Reciprocal Matrix ◽  
Hierarchy Process

Analytic hierarchy process (AHP) is a leading multi-attribute decision-aiding model that is designed to help make better choices when faced with complex decisions involving several dimensions. AHP, which enables qualitative analysis using a combination of subjective and objective information, is a multiple criteria decision analysis approach that uses hierarchical structured pairwise comparisons. One of the drawbacks of AHP is that a pairwise comparison cannot be completed by an actor or stakeholder not fully familiar with all the aspects of the problem. The authors have developed a completion based on a process of linearization that minimizes the matrix distance defined in terms of the Frobenius norm (a strictly convex minimization problem). In this paper, we characterize when an incomplete, positive, and reciprocal matrix can be completed to become a consistent matrix. We show that this characterization reduces the problem to the solution of a linear system of equations—a straightforward procedure. Various properties of such a completion are also developed using graph theory, including explicit calculation formulas. In real decision-making processes, facilitators conducting the study could use these characterizations to accept an incomplete comparison body given by an actor or to encourage the actor to further develop the comparison for the sake of consistency.

scholarly journals The Triads Geometric Consistency Index in AHP-Pairwise Comparison Matrices

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 926 ◽  
Author(s):  
Juan Aguarón ◽  
María Teresa Escobar ◽  
José María Moreno-Jiménez ◽  
Alberto Turón
Keyword(s):  
Pairwise Comparison ◽  
Consistency Index ◽  
Analytic Hierarchy ◽  
Priority Vector ◽  
Decision Tools ◽  
Geometric Consistency ◽  
The Matrix ◽  
Stability Intervals ◽  
Definition Of ◽  
Hierarchy Process

The paper presents the Triads Geometric Consistency Index ( T - G C I ), a measure for evaluating the inconsistency of the pairwise comparison matrices employed in the Analytic Hierarchy Process (AHP). Based on the Saaty’s definition of consistency for AHP, the new measure works directly with triads of the initial judgements, without having to previously calculate the priority vector, and therefore is valid for any prioritisation procedure used in AHP. The T - G C I is an intuitive indicator defined as the average of the log quadratic deviations from the unit of the intensities of all the cycles of length three. Its value coincides with that of the Geometric Consistency Index ( G C I ) and this allows the utilisation of the inconsistency thresholds as well as the properties of the G C I when using the T - G C I . In addition, the decision tools developed for the G C I can be used when working with triads ( T - G C I ), especially the procedure for improving the inconsistency and the consistency stability intervals of the judgements used in group decision making. The paper further includes a study of the computational complexity of both measures ( T - G C I and G C I ) which allows selecting the most appropriate expression, depending on the size of the matrix. Finally, it is proved that the generalisation of the proposed measure to cycles of any length coincides with the T - G C I . It is not therefore necessary to consider cycles of length greater than three, as they are more complex to obtain and the calculation of their associated measure is more difficult.

scholarly journals Morphometric parameters based prioritization of a Mid-Himalayan watershed using fuzzy analytic hierarchy process

2021 ◽  
Vol 280 ◽  
pp. 10004
Author(s):  
Mallika Joshi ◽  
Pankaj Kumar ◽  
Purabi Sarkar
Keyword(s):  
Analytic Hierarchy Process ◽  
Pairwise Comparison ◽  
Decision Support Tool ◽  
Morphometric Parameters ◽  
Fuzzy Analytic Hierarchy Process ◽  
Analytic Hierarchy ◽  
Pairwise Comparison Matrix ◽  
Support Tool ◽  
The Matrix ◽  
Hierarchy Process

Watershed prioritization has become increasingly crucial for managing natural resources, especially the watersheds. A useful decision support tool to provide appropriate weights to different morphological attributes with lineage with soil erosion is required to identify environmentally stressed areas for the watershed resources. This study examines the Western Nayar watershed delineation and further examination of the watershed’s morphometric parameters. The morphometric parameters were quantified under the linear, areal, and relief heads for the watershed. The prioritization of sub-watersheds was done by the fuzzy analytic hierarchy process (FAHP). The study included nine morphometric parameters for forming a pairwise comparison matrix. The fuzzy analytic hierarchy process was employed for assigning the suitable weights to morphometric parameters, and further, these weights are normalized to assign the final ranks to the sub-watershed. In Western Nayar, SW9 got the highest priority, and SW1 was categorized as the least priority. The results were validated by the consistency ratio index, which depends on the matrix consistency index’s size that should be less than 10%. The consistency index of the present study was found to be 2%.

DETERMINING THE DESALINATION UNIT COMPOSITION FOR COSTAL AREAS AND SMALL ISLANDS USE ANALYTIC HIERARCHY PROCESS

2017 ◽  
Vol 9 (1) ◽  
Author(s):  
Imam Setiadi ◽  
Dinda Rita K. Hartaja
Keyword(s):  
Analytic Hierarchy Process ◽  
Energy Demand ◽  
Pairwise Comparison ◽  
Small Islands ◽  
Ahp Method ◽  
Analytic Hierarchy ◽  
Comparison Research ◽  
Hierarchy Process ◽  
Selection Of

Selection of the appropriate composition desalination units can be done with a variety of method approaches, one of the method is the Analytic Hierarchy Process. In determining the desalination unit with AHP method to consider is setting a goal, an alternative criteria and pairwise comparison. Research for the determination of the exact composition of the desalination unit in order to achieve sustainable drinking water suppy in coastal areas and small islands has been conducted. The results of the study are as follows, the energy demand of 50.83%, operator costs of 26.64%, maintenance costs of 14.13% and chemical requirement 8.4%. For an alternative composition desalination unit of RO 10 m3 / day is the best alternative composition with value of 59.61%, the composition of the next alternative is RO 20 m3/ day of 30.40% and the last alternative of the desalination unit composition is RO 120 m3/ day of 09.99%.Key words : Desalination, Mukti Stage Flash Composition, AHP

On Moderate Fuzzy Analytic Hierarchy Process Pairwise Comparison Model With Subcriteria

2016 ◽  
Vol 2 (3) ◽  
pp. 54
Author(s):  
G. Marimuthu ◽  
G. Ramesh
Keyword(s):  
Decision Making ◽  
Analytic Hierarchy Process ◽  
Fuzzy Ahp ◽  
Pairwise Comparison ◽  
Decision Models ◽  
Fuzzy Analytic Hierarchy Process ◽  
Analysis Method ◽  
Analytic Hierarchy ◽  
Comparison Model ◽  
Hierarchy Process

Decisions usually involve the getting the best solution, selecting the suitable experiments, most appropriate judgments, taking the quality results etc., using some techniques.  Every decision making can be considered as the choice from the set of alternatives based on a set of criteria.  The fuzzy analytic hierarchy process is a multi-criteria decision making and is dealing with decision making problems through pairwise comparisons mode [10].  The weight vectors from this comparison model are obtained by using extent analysis method.  This paper concern with an alternate method of finding the weight vectors from the original fuzzy AHP decision model (moderate fuzzy AHP model), that has the same rank as obtained in original fuzzy AHP and ideal fuzzy AHP decision models.

JUDGMENT NUMBER REDUCTION: AN ISSUE IN THE ANALYTIC HIERARCHY PROCESS

2010 ◽  
Vol 09 (01) ◽  
pp. 175-189 ◽  
Author(s):  
LONG-TING WU ◽  
XIA CUI ◽  
RU-WEI DAI
Keyword(s):  
Decision Making ◽  
Analytic Hierarchy Process ◽  
Pairwise Comparison ◽  
Random Variable ◽  
Analytic Hierarchy ◽  
Priority Vector ◽  
Judgment Error ◽  
Scale Error ◽  
The Analytic Hierarchy Process ◽  
Hierarchy Process

The Analytic Hierarchy Process (AHP) uses pairwise comparison to evaluate alternatives' advantages to a certain criterion. For decision-making problem with many different criteria and alternatives, pairwise comparison causes a prolonged decision-making period and rises fatigue in decision-makers' mentality. A question of practical value is if there exists a way to reduce judgment number and what influence the reduction will have on the overall evaluation of alternative ratings. To answer this question, we introduce scale error and judgment error into AHP judgment matrix. By expanding the scales defined in the AHP, scale error is eliminated. Taking judgment error as random variable, a new estimator to calculate priority vector is presented. In the end, an example is proved to show lowering judgment number will increase the probability of larger errors appearing in priority vector computation.

scholarly journals PENERAPAN METODE FUZZY AHP DALAM PENENTUAN SEKTOR YANG BERPENGARUH TERHADAP PEREKONOMIAN PROVINSI BALI

2016 ◽  
Vol 5 (2) ◽  
pp. 59
Author(s):  
TJOKORDA GDE AGUNG FRISKA ADNYANA ◽  
G. K. GANDHIADI ◽  
DESAK PUTU EKA NILAKUSMAWATI
Keyword(s):  
Manufacturing Industry ◽  
Fuzzy Ahp ◽  
Weighted Average ◽  
Pairwise Comparison ◽  
Primary Group ◽  
Tertiary Sector ◽  
Analytic Hierarchy ◽  
Consistency Ratio ◽  
Analytic Hierarchy Process Method ◽  
Hierarchy Process

The aim of this research is to apply Fuzzy Analytic Hierarchy Process method to determine the dominant sectors of economy Bali Province. This research used survey data from respondents who understand about economy of Bali Province and Gross Regional Domestic Product (GRDP). The research variables consist of criteria and sub-criteria. The criteria consist of primary sector, secondary, and tertiary, while the sub-criteria consist of 9 sectors of the GRDP. Data processing is done by calculating the weighted average and arrange them into pairwise comparison matrices. Furthermore, the consistency ratio is checked. If consistency ratio less than 0.100 (), the elements of matrices is changed into triangular fuzzy scale and processed with synthetic extent to get the priority. Based on the result of research, the economy of Bali are dominated by tertiary sector in sector group, agriculture and forestry in sub-primary group, manufacturing industry in sub-secondary group, and trade, hotel and restaurant in sub-tertiary group.

scholarly journals Evaluating and ranking the circular supply chain implementation enablers

2021 ◽  
Author(s):  
Swapnil Lahane ◽  
◽  
Himanshu Prajapati ◽  
Ravi Kant ◽  
◽  
...  
Keyword(s):  
Supply Chain ◽  
Manufacturing Industry ◽  
Global Climate ◽  
Pairwise Comparison ◽  
Decision Makers ◽  
Analytic Hierarchy ◽  
Effective Implementation ◽  
Manufacturing Organizations ◽  
Scarcity Of Resources ◽  
Hierarchy Process

This research aims to evaluate and rank the Circular Supply Chain (CSC) enablers for the effective implementation of CSC management. From the literature and input received from industrial experts, 30 CSC enablers are identified. Further, the selected CSC enablers are classified into seven main categories. This research employs the Pythagorean fuzzy analytic hierarchy process (PF-AHP) technique to prioritize the selected CSC enablers based on pairwise comparison of criteria and sub-criteria. The effectiveness of the proposed method is illustrated with the help of responses received from an Indian manufacturing industry. The result reveals that global climate pressure and ecological scarcity of resources is the most significant enabler and the environment management certifications and systems is the least important enabler. The result of the present study will assist the practitioners / decision makers in implementing CSC in manufacturing organizations by adopting the ranking obtained in a systematic way.

scholarly journals An Approach for Generating Weights Using the Pairwise Comparison Matrix

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Zaher Sepehrian ◽  
Sahar Khoshfetrat ◽  
Said Ebadi
Keyword(s):  
Pairwise Comparison ◽  
Decision Criterion ◽  
Data Envelopment ◽  
Decision Criteria ◽  
Analytic Hierarchy ◽  
Pairwise Comparison Matrix ◽  
Proposed Model ◽  
Assurance Region ◽  
Potential Applications ◽  
Hierarchy Process

Data envelopment analysis (DEA) has been used for obtaining weights for the analytic hierarchy process (AHP), an approach known as DEAHP. This method sometimes identifies more than one decision criterion or alternative as DEAHP-efficient. To overcome this problem, this paper proposes a new approach that not only generates appropriate weights for the decision criteria or alternatives, but also differentiates between DEAHP-efficient decision criteria or alternatives. To this end, we propose a DEA model with an assurance region and a cross-weight model that prioritizes decision criteria or alternatives by considering their most unfavorable weights. Two numerical examples are also provided to illustrate the advantages and potential applications of the proposed model.

Comparison of methods of ranking scientists based on the Hirsch and PageRank indices: contradictory situations

2021 ◽  
pp. 39-44
Author(s):  
Hryhorii Hnatiienko ◽  
Oleksiy Oletsky
Keyword(s):  
Analytic Hierarchy Process ◽  
Pairwise Comparison ◽  
The Other ◽  
Publication Activity ◽  
Comparison Of Methods ◽  
Automated Systems ◽  
Analytic Hierarchy ◽  
H Index ◽  
Scientometric Indicators ◽  
Hierarchy Process

Experiments aimed at comparing different methods of estimating and ranking scientists and researchers on the base of their publication activity are reported. Scientometric indicators based on h-index and PageRank are being compared. For such a comparison, a graph of citations represented by a matrix was applied. An example when different methods lead to opposite results was described. For example, authors having the best PageRank-based estimations may have the least h-indices. Such a situation is possible when a high-cited author managed to obtain a key result cited by all the other authors but this author has few papers. A comparison with methods of expert estimations was carried out, which appears to be very useful for building automated systems combining various methods of algorithmic estimating and ranking. The Analytic Hierarchy Process was applied. For building pairwise comparison matrices, transitive scales with a parameter representing how much times the next level of advantage is bigger than the previous one were harnessed.

The Problem of Adequacy of the Analytic Hierarchy Process

2020 ◽  
Vol 10 (4) ◽  
pp. 79-87
Author(s):  
V.M. Romanchuk
Keyword(s):  
Analytic Hierarchy Process ◽  
Pairwise Comparison ◽  
Measurement Procedure ◽  
Ahp Method ◽  
Calculation Algorithm ◽  
Analytic Hierarchy ◽  
Rating Method ◽  
Simple Measurement ◽  
The Analytic Hierarchy Process ◽  
Hierarchy Process

The Analytic hierarchy process (AHP) is a popular method for solving multi-criteria problems. However, the problem of the adequacy of the AHP method is not solved. Opponents of the Analytic hierarchy process believe that the AHP as a whole is erroneous and cannot be applied in practice. Proponents of the method believe that the disadvantages of the method are compensated by a simple measurement procedure. In this paper, a modification of the AHP method is proposed. A mathematical model of measurement is formulated, which contains a built-in mechanism for checking adequacy. moreover, the measurement method is preserved, and the calculation algorithm becomes even simpler. The fact is that the Analytic hierarchy process is based on the assumption that the scale of relations can be obtained by pairwise comparison using numerical judgments based on the absolute scale of numbers. Fechner’s psychophysical law is considered as a justification for the existence of the scale of relations. But there are not one, but two psychophysical laws. The existence of two psychophysical laws is a problem of psychophysics. This problem can be solved by the rating method. To overcome the disadvantages of the Analytic hierarchy process, it is also proposed to use the rating method. The use of the rating method makes it possible to use the fundamental scale of the AHP method. As an example, the problem is solved using the traditional AHP scale.

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