A constrained non-linear optimization model for fuzzy pairwise comparison matrices using teaching learning based optimization

Non-linear optimization models have been recently proposed to derive crisp weights from fuzzy pairwise comparison matrices. In this paper, a TLBO (Teaching Learning Based Optimization) based solution is presented for solving an optimization model as a system of non-linear equations to derive crisp weights from fuzzy pairwise comparison matrices in AHP (Analytic Hierarchy Process). This fuzzy-AHP method is named as TLBO-1. It has been found that TLBO-1 can lead to inconsistent or less consistent weights. To solve the problem of inconsistent weights, a new constrained non-linear optimization model is proposed in this paper. This model is based on the min-max approach for fuzzy pairwise comparison ratios of weights. TLBO is again used to solve this optimization model, and crisp weights are derived. This fuzzy AHP method is named as TLBO-2. The effectiveness of the proposed model is illustrated by three examples. For each example, the consistency of the derived crisp weights is compared with other optimization models. The results show that the TLBO-2 method can derive more consistent weights for the fuzzy AHP based Multi-Criteria Decision Making (MCDM) systems as compared to the other optimization models.     

A linear programming method for generating the most favorable weights from a pairwise comparison matrix

This paper proposes a linear programming method for generating the most favorable weights (LP-GFW) from pairwise comparison matrices, which incorporates the variable weight concept of data envelopment analysis (DEA) into the priority scheme of the analytic hierarchy process (AHP) to generate the most favorable weights for the underlying criteria and alternatives on the basis of a crisp pairwise comparison matrix. The proposed LP-GFW method can generate precise weights for perfectly consistent pairwise comparison matrices and approximate weights for inconsistent pairwise comparison matrices, which are not too far from Saaty's principal right eigenvector weights. The issue of aggregation of local most favorable weights and rank preservation methods is also discussed. Four numerical examples are examined using the LP-GFW method to illustrate its potential applications and significant advantages over some existing priority methods.     

A fuzzy AHP approach to the determination of importance weights of customer requirements in quality function deployment

Quality function deployment (QFD) is an important tool in product planning that could contribute to increase in customer satisfaction and shorten product design and development time. During the QFD process, determination of the importance weights of customer requirements is a crucial and essential step. The analytic hierarchy process (AHP) has been used in weighting the importance. However, due to the vagueness and uncertainty existing in the importance attributed to judgement of customer requirements, the crisp pairwise comparison in the conventional AHP seems to be insufficient and imprecise to capture the degree of importance of customer requirements. In this paper, fuzzy number is introduced in the pairwise comparison of AHP. An AHP based on fuzzy scales is proposed to determine the importance weights of customer requirements. The new approach can improve the imprecise ranking of customer requirements which is based on the conventional AHP. Finally, an example of bicycle splashguard design is used to illustrate the proposed approach.     

The analytic hierarchy process: a note on an approach to sensitivity which preserves rank order

Within the framework of the AHP as it applies to multicriteria decisions, it is frequently the case that decision makers are certain about the rank order of the objects for a particular pairwise comparison matrix but uncertain about the precise numerical weights that the AHP produces for that matrix. This uncertainty translates directly into uncertainty about whether the best alternative obtained from the AHP is actually the best alternative. However, if the weights of an AHP pairwise comparison matrix can be varied in a way that preserves the rank order of the objects, and at the same time, this perturbation does not result in the best alternative changing, then the decision maker is typically much more confident about what the AHP recommends. In this paper, I detail a simple approach to sensitivity within the AHP which preserves the rank order of the objects.Scope and purposeIn the author's experience with AHP as a multicriteria decision tool, it is frequently the case that decision makers (DMs) are quite certain about the rank order of the objects for a particular pairwise comparison matrix (PCM) but uncertain about the precise numerical weights that the AHP produces for that matrix. This uncertainty translates directly into uncertainty about whether the best alternative obtained from the AHP is actually the best alternative. However, if the weights of a PCM can be varied in a way that preserves the rank order of the objects for that matrix, and at the same time, this perturbation does not result in the best alternative overall changing, then the DM is typically much more confident about what the AHP recommends. In this paper I detail such an approach to sensitivity for the AHP.     

区间互补判断矩阵可接受一致性

对区间互补判断矩阵的一致性进行研究,提出一种新的可接受一致性定义,将不满足可接受一致性的矩阵较容易地修正为可接受一致性矩阵.基于凸组合方法,一族明晰数互补判断矩阵的权重向量可被用来求取可接受一致性区间互补判断矩阵的区间权重,并提出了求取可接受一致性区间互补判断矩阵区间权重向量的算法.数值例子显示了所提出的可接受一致性定义以及算法的可行性和有效性.     

改进的AHP算法在Web集群系统负载均衡中的应用

判断矩阵的建立和调整是AHP算法的关键。对Satty的1~9标度方法进行改进,引入3标度法建立判断矩阵。从正互反矩阵充分必要条件出发,提出判断矩阵不一致时的调整方法,以提高判断矩阵的一致性和减少计算量。该方法应用于Web集群系统负载均衡中,确定影响服务器负载的四大类参数的权重。实验表明,改进的AHP算法在判断矩阵不一致时,在尊重原始数据的基础上,调整判断矩阵的计算量减少,判断矩阵的一致性更好,同时可以提高负载均衡系统的性能。     

基于试探算法的AHP评价方法研究

在对土地集约利用评价过程中,存在着评价专家选择的问题以及AHP打分过程中相对矩阵的约束问题。针对这些问题,应用MDS和[K]均值聚类,筛选不同倾向的专家;利用改进的判断矩阵生成算法,对专家的判断矩阵进行一致性检验,解决可能存在的成对比较项目不一致的问题,为构建评价指标体系提供有效的支持。     

Effects of increasing fuzziness on analytic hierarchy process for spatial multicriteria decision analysis

Multicriteria decision analysis (MCDA) involves techniques which relatively recently have received great increase in interest for their capabilities of solving spatial decision problems. One of the most frequently used techniques of MCDA is Analytic Hierarchy Process (AHP). In the AHP, decision-makers make pairwise comparisons between different criteria to obtain values of their relative importance. The AHP initially only dealt with crisp numbers or exact values in the pairwise comparisons, but later it has been modified and adapted to also consider fuzzy values. It is necessary to empirically validate the ability of the fuzzified AHP for solving spatial problems. Further, the effects of different levels of fuzzification on the method have to be studied. In the context of a hypothetical GIS-based decision-making problem of locating a dam in Costa Rica using real-world data, this paper illustrates and compares the effects of increasing levels of uncertainty exemplified through different levels of fuzzification of the AHP. Practical comparison of the methods in this work, in accordance with the theoretical research, revealed that by increasing the level of uncertainty or fuzziness in the fuzzy AHP, differences between results of the conventional and fuzzy AHPs become more significant. These differences in the results of the methods may affect the final decisions in decision-making processes. This study concludes that the AHP is sensitive to the level of fuzzification and decision-makers should be aware of this sensitivity while using the fuzzy AHP. Furthermore, the methodology described may serve as a guideline on how to perform a sensitivity analysis in spatial MCDA. Depending on the character of criteria weights, i.e. the degree of fuzzification, and its impact on the results of a selected decision rule (e.g. AHP), the results from a fuzzy analysis may be used to produce sensitivity estimates for crisp AHP MCDA methods.     

Applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy AHP

The lack of consistency in decision making can lead to inconsistent conclusions. In fuzzy analytic hierarchy process (fuzzy AHP) method, it is difficult to ensure a consistent pairwise comparison. Furthermore, establishing a pairwise comparison matrix requires judgments for a level with n criteria (alternatives). The number of comparisons increases as the number of criteria increases. Therefore, the decision makers judgments will most likely be inconsistent. To alleviate inconsistencies, this study applies fuzzy linguistic preference relations (Fuzzy LinPreRa) to construct a pairwise comparison matrix with additive reciprocal property and consistency. In this study, the fuzzy AHP method is reviewed, and then the Fuzzy LinPreRa method is proposed. Finally, the presented method is applied to the example addressed by Kahraman et al. [C. Kahraman, D. Ruan, I. Do?an, Fuzzy group decision making for facility location selection, Information Sciences 157 (2003) 135-153]. This study reveals that the proposed method yields consistent decision rankings from only n − 1 pairwise comparisons, which is the same result as in Kahraman et al. research. The presented fuzzy linguistic preference relations method is an easy and practical way to provide a mechanism for improving consistency in fuzzy AHP method.     

Measuring the success possibility of implementing ERP by utilizing the Incomplete Linguistic Preference Relations

This paper applies an analytic hierarchical prediction model based on the Multi-Criteria Decision Making with Incomplete Linguistic Preference Relations (InLinPreRa) to help the organizations become aware of the essential factors affecting the Enterprise Resource Planning (ERP), as well as identify the actions necessary before implementing ERP. The subjectivity and vagueness in the prediction procedures are dealt with linguistic variables quantified in an interval [−t, t]. Then predicted success/failure values are obtained to enable organizations to decide whether to initiate ERP, inhibit adoption or take remedial actions to increase the success possibility of ERP. Pairwise comparisons are used to determine the priority weights of influential factors, and the possible occurrence ratings of success or failure outcome amongst decision makers. There are not any inconsistency occurred in this procedures because this proposed approach allows every decision expert to choose an explicit criterion or alternative for the without restriction. When there are n criteria in a decision matrix, only n − 1 times of pairwise comparisons are taken. This approach not only improves the efficiency of pairwise comparison compared with the traditional AHP, but also avoids the checking the consistency of linguistic preference relation when the decision makers undertake the pairwise comparison processes.     

效能评估中的改进熵值法赋权研究

针对熵值法赋权时当某个指标的离散程度太大时,该指标的权重会过大这一问题,借鉴层次分析法（AHP）的赋权思路,对熵值法进行了改进,通过对指标差异性系数进行两两比较获得的判断矩阵来求解指标权重。详细给出了保持指标信息熵不变的指标初始数据的规范化方法,计算从AHP赋权法的1~9标度到改进熵值法的1~9标度映射值的步骤,以及构造基于信息熵的判断矩阵并求解指标权重的方法。分别利用熵值法和改进的熵值法对战斗机空空作战能力的六项指标赋权,对战斗机空空作战能力进行评估。结果表明改进熵值法弥补了熵值法赋权时单一指标权重可能过大的不足,赋权结果更加合理,空空作战能力评估结果也更为准确。     

一种调整AHP不一致判断矩阵的优化方法

针对AHP中不一致判断矩阵,提出了一种新的调整方法. 通过将AHP不一致判断矩阵的调整问题等价转化为一个带约束条件的优化问题,能够保证判断矩阵满足一致性要求条件下,使得调整后判断矩阵与原始判断矩阵的差异程度最小. 给出了决策容许区间的概念,使得元素的变动值在规定的范围内,同时,采取逐渐增大扰动变量的方法控制决策容许区间,以保证最小的改动实现一致性要求,并设计了改进的模式搜索算法求解优化问题. 最后,通过算例说明了方法的可行性. 该方法与现有方法相比,不仅具有直观性,易于计算机编程实现,而且能够真正做到以最小的改动量实现判断矩阵满足一致性要求.     

基于AHP的模糊线性规划求解方法

对于基于AHP的多准则分析过程,存在不一致区间判断的复杂评估问题.通过有下限和上限的区间数表示元素之间的比较比率,构造模糊约束集合矩阵,引入模糊集的隶属度函数表示对各种优先权矢量的满意程度,利用线性规划求解具有最大满意度的优先权矢量,得出候选者的总体优先顺序,并举例说明了应用该方法的计算过程.     

基于AHP的模糊线性规划在区域经济评估中的应用

对于区域经济评估,在AHP框架下,存在不一致区间判断的情况。通过运用有上、下限的区间数表示元素之间的比较比率,构造模糊约束集合矩阵,引入模糊集的隶属度函数表示对各种优先权矢量的满意程度,利用线性规划求解具有最大满意度的优先权矢量,得出候选者的总体优先顺序,并举例说明了应用该方法的计算过程。     

A parallel approach to the analytic hierarchy process decision support tool

The Multiple Criteria Decision Aiding methods dedicated to discrete problems follow different philosophies and strategies for selecting, clustering or ranking alternatives. This work presents a tool using one such method—the Analytic Hierarchy Process (AHP). The Decision Maker (DM) can structure his criteria as a hierarchy tree having the alternatives as leaf nodes. The DM must then build matrices for each node by performing pairwise comparisons between its children. The AHP finds the weights of each child concerning the parent criterion by calculating the elements of the eigenvector corresponding to the maximum eigenvalue of the comparison matrix. Weights are then combined in order to obtain the influence of each alternative on the top of the hierarchy. A DM expects that a Decision Support Tool works faster than he/she does. In order to achieve speed a parallel approach was developed. Parallel implementations described in this work follow different message-passing strategies and capitalise on the fact that the vector of weights for each matrix can be calculated independently. The authors used a network of four Inmos Transputers. Research will focus on finding which implementation will run faster and how the DMs options affect the speedups obtainable.     

Linear programming models for estimating weights in the analytic hierarchy process

We present an approach based on linear programming (LP) that estimates the weights for a pairwise comparison matrix generated within the framework of the analytic hierarchy process. Our approach makes sense for a number of reasons, which we discuss. We apply our LP approach to several sample problems and compare our results to those produced by other, widely used methods. In addition, we extend our linear program to include applications where the pairwise comparison matrix is constructed from interval judgments.     

基于模糊AHP的系统仿真可信度评估方法

该文讨论用模糊层次分析法(fuzzy AHP)计算影响系统仿真可信度评估的各因素的权重，fuzzy AHP基于模糊理论，用三角模糊数表示两两判断矩阵。通过比较第1个对象满足m个目标的综合程度计算权重向量。filzzy AHP用法考虑了人们对复杂事物判断的模糊性，能够更好地克服人为主观判断、偏好给决策带来的影响，因而更具有合理性。最后的应用结果表明该方法科学合理，简单实用，具有一定的工程应用价值。     

Interval regression analysis by quadratic programming approach

When we use linear programming in possibilistic regression analysis, some coefficients tend to become crisp because of the characteristic of linear programming. On the other hand, a quadratic programming approach gives more diverse spread coefficients than a linear programming one. Therefore, to overcome the crisp characteristic of linear programming, we propose interval regression analysis based on a quadratic programming approach. Another advantage of adopting a quadratic programming approach is to be able to integrate both the property of central tendency in least squares and the possibilistic property in fuzzy regression. By changing the weights of the quadratic function, we can analyze the given data from different viewpoints. For data with crisp inputs and interval outputs, the possibility and necessity models can be considered. Therefore, the unified quadratic programming approach obtaining the possibility and necessity regression models simultaneously is proposed. Even though there always exist possibility estimation models, the existence of necessity estimation models is not guaranteed if we fail to assume a proper function fitting to the given data as a regression model. Thus, we consider polynomials as regression models since any curve can be represented by the polynomial approximation. Using polynomials, we discuss how to obtain approximation models which fit well to the given data where the measure of fitness is newly defined to gauge the similarity between the possibility and the necessity models. Furthermore, from the obtained possibility and necessity regression models, a trapezoidal fuzzy output can be constructed     

A consensus approach for synthesizing the elements of comparison matrix in the Analytic Hierarchy Process

The Analytic Hierarchy Process (AHP) has been widely utilized to solve multicriteria decision-making problems in synthesizing conflict opinions. Normally, AHP uses the geometric averaging approach to synthesize preference weights determined by decision-makers. This approach has been criticized by many researches since synthesis weight may not reach a consensus. To make the synthesis acceptable to all decision-makers, the study proposes a computer-aided approach to achieve a compromise for all the elements in the comparison matrix while implementing AHP. Accordingly, decision-makers can conveniently exchange trustful information, which is generated by the embedded genetic algorithm, sensitivity analysis and similarity measure of the judgements done by decision-makers. Consequently, the consensus of all the elements in the comparison matrix can be obtained through such an innovative approach to resolve the disparity of judgements within decisionmakers.     

Modifying inconsistent comparison matrix in analytic hierarchy process: A heuristic approach

Test of consistency is a critical step in the AHP methodology. When a pairwise comparison matrix fails to satisfy the consistency requirement, a decision maker needs to make revisions. To aid the decision maker's revising process, several approaches identify changes to the consistency requirement with respect to changes to a single entry in the original inconsistent matrix [P.T. Harker, Derivatives of the Perron root of a positive reciprocal matrix: with application to the analytic hierarchy process, Applied Mathematics and Computation, 22, 217–232 (1987); T.L. Saaty, Decision-making with the AHP: Why is the principal eigenvector necessary, European Journal of Operational Research, 145, 85–91 (2003)]. Instead of revising single entries, Xu and Wei [Z. Xu and C. Wei, A consistency improving method in the analytic hierarchy process, European Journal of Operational Research, V.116, 443–449 (1999)] derived a consistent matrix by an auto-adaptive process based on the original inconsistent matrix. In this paper, we develop a heuristic approach that auto-generates a consistent matrix based on the original inconsistent matrix. Expressing the inconsistent matrix in terms of a deviation matrix, an iterative process adjusts the deviation matrix to improve the consistency ratio, while preserving most of the original comparison information. We show that the proposed method is able to preserve more original comparison information than Xu and Wei [Z. Xu and C. Wei, A consistency improving method in the analytic hierarchy process, European Journal of Operational Research, V.116, 443–449 (1999)]. It is also shown that the heuristic approach can be used to examine the effects of revising a sub-bloc as well as revising a single entry of the original matrix.