scholarly journals On a Maximum Eigenvalue of Third-Order Pairwise Comparison Matrix in Analytic Hierarchy Process and Convergence of Newton’s Method

2021 ◽  
Vol 2 (3) ◽  
Author(s):  
Shunsuke Shiraishi ◽  
Tsuneshi Obata
Keyword(s):  
Analytic Hierarchy Process ◽  
Real Number ◽  
Pairwise Comparison ◽  
Maximum Magnitude ◽  
Third Order ◽  
Analytic Hierarchy ◽  
Pairwise Comparison Matrix ◽  
Comparison Matrix ◽  
The Third ◽  
Hierarchy Process

AbstractNowadays, the analytic hierarchy process is an established method of multiple criteria decision making in the field of Operations Research. Pairwise comparison matrix plays a crucial role in the analytic hierarchy process. The principal (maximum magnitude) eigenvalue of the pairwise comparison matrix can be utilized for measuring the consistency of the decision maker’s judgment. The simple transformation of the maximum magnitude eigenvalue is known to be Saaty’s consistency index. In this short note, we shed light on the characteristic polynomial of a pairwise comparison matrix of third order. We will show that the only real-number root of the characteristic equation is the maximum magnitude eigenvalue of the third-order pairwise comparison matrix. The unique real-number root appears in the area where it is greater than 3, which is equal to the order of the matrix. By applying usual Newton’s method to the characteristic polynomial of the third-order pairwise comparison matrix, we see that the sequence generated from the initial value of 3 always converges to the maximum magnitude eigenvalue.

Electronic Market Credit Risk Evaluation for Agricultural Products Based on Analytic Hierarchy Process

2011 ◽  
Vol 99-100 ◽  
pp. 852-856
Author(s):  
You Zhu Li ◽  
De Hua He
Keyword(s):  
Analytic Hierarchy Process ◽  
Credit Risk ◽  
Risk Evaluation ◽  
Pairwise Comparison ◽  
Agricultural Products ◽  
Electronic Market ◽  
Analytic Hierarchy ◽  
Pairwise Comparison Matrix ◽  
Comparison Matrix ◽  
Hierarchy Process

In the study, electronic market credit risk evaluation for agricultural products based on analytic hierarchy process is proposed.Firstly, the evaluation indexes are analyzed and the hierarchic tree is formulated based on the evaluation indexes.Then, pairwise comparison matrix is established,and the consistency of discriminant matrix is judged.When the consistency of discriminant matrix is satisfied,the weight vector of the indexes which are used to establish the pairwise comparison matrix are obtained. And weight of each index is obtained.Finally,final decision making is obtained. The experimental results show that the evaluation of electronic market credit risk evaluation for agricultural products based on analytic hierarchy process is effective.

scholarly journals APPLYING THE ANALYTIC HIERARCHY PROCESS TO OIL SANDS ENVIRONMENTAL COMPLIANCE RISK MANAGEMENT

2016 ◽  
Vol 8 (1) ◽  
Author(s):  
Izak Johannes Roux ◽  
Dr. Christos Makrigeorgis
Keyword(s):  
Analytic Hierarchy Process ◽  
Oil Sands ◽  
Pairwise Comparison ◽  
Environmental Compliance ◽  
Analytic Hierarchy ◽  
Pairwise Comparison Matrix ◽  
Comparison Matrix ◽  
Risk Criteria ◽  
The Analytic Hierarchy Process ◽  
Hierarchy Process

<p>In 2013, oil companies in Alberta, Canada invested $32 billion in new oil-sands projects.  Despite the size of this investment, there is a demonstrable deficiency in the uniformity and understanding of environmental legislation requirements that translate into increased project compliance risks. In this paper, we applied the Analytic Hierarchy Process (AHP) to develop a priority list of environmental regulatory compliance risk criteria for oil-sands projects.  AHP belongs to the family of multicriteria decision-making (MCDM) techniques that utilizes a pairwise comparison matrix solicited from subject matter experts (SMEs) in the field as input.  The overall methodology itself consisted of 4 phases: (1) identification of the initial list of N potential environmental compliance risk criteria and verification of these criteria via a pilot survey; (2) formation of a pairwise comparison survey in the form of an N(N-1)/2 comparison matrix based on the verified criteria; (3) administration of the pairwise comparison matrix to a sample of 16 industry-specific SME’s; and (4) the application of the AHP method using SuperDecisions as a tool on the collected sample to rank the identified risk criteria. Our demonstrated results can potentially inform Alberta oil sands industry leaders about the ranking and utility of specific compliance risks as understood by experts and enable a more focused environmental compliance action to help increase legislative and public trust.</p>

Materials selection criteria weighting method using analytic hierarchy process (AHP) with simplest questionnaire and modifying method of inconsistent pairwise comparison matrix

2021 ◽  
pp. 146442072110399
Author(s):  
Won-Chol Yang ◽  
Jae-Bok Ri ◽  
Ji-Yon Yang ◽  
Ju-Song Kim
Keyword(s):  
Analytic Hierarchy Process ◽  
Selection Criteria ◽  
Pairwise Comparison ◽  
Hot Water ◽  
Materials Selection ◽  
Analytic Hierarchy ◽  
Pairwise Comparison Matrix ◽  
Comparison Matrix ◽  
The Analytic Hierarchy Process ◽  
Hierarchy Process

The analytic hierarchy process has been widely used to determine subjective weights of materials selection criteria in materials selection using multi-criteria decision-making. However, the analytic hierarchy process has some drawbacks: it is difficult to construct a pairwise comparison matrix and meet the consistency requirement. First, we propose a new simplest questionnaire to perform the pairwise comparison without confusion, conventionally and easily. Next, we propose an improved modifying method for inconsistent pairwise comparison matrix according to the following principles: (1) the elements of the reconstructed pairwise comparison matrix should be nine-point scales, (2) the number and modifying the amount of the modified elements should be as small as possible and (3) the deviation between the original and reconstructed pairwise comparison matrixes should be as small as possible. The outline of the proposed modifying method is as follows: (1) calculate the consistency ration decrements of all the pairwise comparison matrixes reconstructed by modifying every element of the original pairwise comparison matrix to the lower and upper adjacent nine-point scales and (2) find the element with the maximum consistency ratio decrement and modify it to the lower or upper adjacent scale. To illustrate the effectiveness, we apply the proposed methods to determine the criteria weights for selecting the best phase change material used in a solar domestic hot water system, and apply the proposed modifying method to some examples from the published papers, and compare the performances with some previous methods. The simplest questionnaire and improved modifying method help materials designers and engineers to apply the analytic hierarchy process method in materials design and optimization problems, much more actively.

Evaluation for Enterprise Marketing Performance Based on Analytic Hierarchy Process

2012 ◽  
Vol 433-440 ◽  
pp. 2109-2113
Author(s):  
Xiao Zhang
Keyword(s):  
Analytic Hierarchy Process ◽  
Marketing Strategy ◽  
Pairwise Comparison ◽  
Relative Importance ◽  
Analytic Hierarchy ◽  
Pairwise Comparison Matrix ◽  
Comparison Matrix ◽  
Marketing Performance ◽  
Global Weight ◽  
Hierarchy Process

Evaluation of enterprise marketing performance has a great importance for enterprise to formulate marketing strategy and carry out marketing activity.Evaluation for enterprise marketing performance based on analytic hierarchy process is proposed in the paper.Analytic hierarchy process is used to perform multi-criteria decision analysis in order to determine the relative importance in the decision matrix.The pairwise comparison matrix by using the pairwise comparison of the indexes in same layer is established.Local weight and global weight are computated by using the pairwise comparison matrix. Marketing condition of a certain enterprise is applied to evaluate for enterprise marketing performance. It can be seen that enterprise marketing performance based on analytic hierarchy process is effective.

Improved Consistency Ratio for Pairwise Comparison Matrix in Analytic Hierarchy Processes

2016 ◽  
Vol 33 (03) ◽  
pp. 1650020
Author(s):  
L. N. Pradeep Kumar Rallabandi ◽  
Ravindranath Vandrangi ◽  
Subba Rao Rachakonda
Keyword(s):  
Error Function ◽  
Pairwise Comparison ◽  
Large Set ◽  
Simulation Experiments ◽  
Analytic Hierarchy ◽  
Chi Square ◽  
Pairwise Comparison Matrix ◽  
Comparison Matrix ◽  
Hierarchy Process ◽  
Original Information

The analytical hierarchy process (AHP) uses pairwise comparison matrix (PCM) to rank a known set of alternatives. Sometimes the comparisons made by the experts may be inconsistent which results in incorrect weights and rankings for the AHP. In this paper, a method is proposed which identifies inconsistent elements in a PCM and revises them iteratively until the inconsistency is reduced to an acceptable level. An error function similar to chi-square is used to identify the inconsistent elements which are revised with suitable values. The method is illustrated with some numerical examples mentioned in the literature and a comparative study of the results in terms of deviation from the PCM and preservation of original information is taken up. Monte Carlo simulation experiments over a large set of random matrices indicate that the proposed method converges for the moderately inconsistent matrices.

scholarly journals Ranking All DEA-Efficient DMUs Based on Cross Efficiency and Analytic Hierarchy Process Methods

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Dariush Akbarian
Keyword(s):  
Analytic Hierarchy Process ◽  
Pairwise Comparison ◽  
Analytic Hierarchy ◽  
Pairwise Comparison Matrix ◽  
Dea Models ◽  
Dea Efficiency ◽  
One Step ◽  
The Cross ◽  
The One ◽  
Hierarchy Process

The aim of this paper is to present an original approach for ranking of DEA-efficient DMUs based on the cross efficiency and analytic hierarchy process (AHP) methods. The approach includes two basic stages. In the first stage using DEA models the cross efficiency value of each DEA-efficiency DMU is specified. In the second stage, the pairwise comparison matrix generated in the first stage is utilized to rank scale of the units via the one-step process of AHP. The advantage of this proposed method is its capability of ranking extreme and nonextreme DEA-efficient DMUs. The numerical examples are presented in this paper and we compare our approach with some other approaches.

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