A novel meta-heuristic based method for deriving priorities from fuzzy pairwise comparison judgments

This paper proposes a new method to derive the priority vector from fuzzy pairwise comparison matrices. Unlike several known methods, the proposed method derives crisp weights from consistent and inconsistent fuzzy comparison matrices. Therefore, the crisp weights obviate the need of additional aggregation and ranking procedures. To derive the priority vector, a Modified Fuzzy Logarithmic Least Square Model (MFLLSM) is proposed. In order to solve the MFLLSM, a framework based on genetic algorithm is proposed. In the proposed framework, a heuristic algorithm of population initialization, a heuristic algorithm for simulating fuzzy numbers and a heuristic algorithm of fitness evaluation are proposed.The solution of the prioritization problem requires finding priorities such that their ratio approximately satisfies the initial judgments. Computational results reveal the superiority of the proposed method in comparison with five well known methods of literature from the viewpoint of satisfaction of initial judgments by the obtained priority vector. It is shown by ten different examples that the deviation of the priorities ratio from initial judgments in the proposed method is less than five existing methods of literature. In addition, unlike several methods of literature, the proposed method considers fuzzy judgments represented by both triangular and trapezoidal fuzzy numbers. Furthermore, the proposed method for the first time considers judgments represented by triangular shaped fuzzy numbers and trapezoidal shaped fuzzy numbers which are discussed in the paper.     

An optimization framework towards prioritization in fuzzy comparison matrices

Prioritization or eliciting final weights from the pairwise comparison matrices might be a major part of some Multiple Criteria Decision Analysis (MCDA) methods which are based upon these matrices. However, in a fuzzy environment, two other issues, i.e., consistency and reciprocity, are also arisen in such matrices with at least one fuzzy component. In this paper, a single-decision-making optimization model along with two group-decision-making optimization models are developed towards prioritization in fuzzy pairwise comparison matrices. A comprehensive numerical analysis is performed to validate the proposed models. Finally, some conclusions are reported.     

Prioritization of hesitant multiplicative preference relations based on data envelopment analysis for group decision making

Hesitant multiplicative preference relations (HMPRs) are utilized to describe situations where a decision maker gives several possible values by Saaty’s 1-9 scale in pairwise comparison. For further applications of HMPRs, this paper develops two priority methods based on data envelopment analysis (DEA) for group decision making. These methods include self-weight prioritization and the cross-weight prioritization, which are similar to the self-evaluation efficiency and the cross-evaluation efficiency in DEA theory, respectively. We prove that both of them can generate true priority weights for consistent HMPRs. The mechanisms of these proposed methods are illustrated with numerical examples. Also, comparisons with other methods are performed to show the advantages of the proposed methods.

     

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 approximation to the eigenvector method in the analytic hierarchy process

Eigenvector method (EM) is a well-known approach to deriving priorities from pairwise comparison matrices in the analytic hierarchy process (AHP), which requires the solution of a set of nonlinear eigenvalue equations. This paper proposes an approximate solution approach to the EM to facilitate its computation. We refer to the approach as a linear programming approximation to the EM, or LPAEM for short. As the name implies, the LPAEM simplifies the nonlinear eigenvalue equations as a linear programming for solution. It produces true weights for perfectly consistent pairwise comparison matrices. Numerical examples are examined to show the validity and effectiveness of the proposed LPAEM and its significant advantages over a recently developed linear programming method entitled LP-GW-AHP in rank preservation.     

基于多等级方案成对比较的决策方法

为了解决现有的方案两两比较方法一般构建单等级上的比较关系, 且不能同时表达多种不同偏好关系的不足, 提出一种新的基于多等级方案成对比较的决策方法, 构建方案集上基于对称框架的分布式多等级偏好关系. 通过设定各等级的得分值, 计算分布式偏好关系的得分矩阵, 并基于此矩阵构建成对优化模型, 求取各方案的优先权区间, 进而产生决策结果. 将所提出的方法应用于某制造企业云服务供应商的选择问题, 验证了所提出方法的应用性和有效性.

     

Modification of the fuzzy analytic hierarchy process via different ranking methods

The use of fuzzy set theory in the analytic hierarchy process (AHP) has gained popularity in recent years as part of the multiple criteria decision-making (MCDM) process to more realistically reflect human judgment. However, due to the nature of fuzzy calculations, this situation imposes more computational load. The aim of this study is to propose methods for obtaining accurate weights from fuzzy pairwise comparison matrices with the least amount of computational load possible. In this context, two different fuzzy AHP (FAHP) methods based on fuzzy numbers ranking methods have been proposed and these proposed methods are compared with commonly accepted FAHP methods. Magnitude-based fuzzy AHP (MFAHP), which is one of the proposed methods, has outperformed all other methods according to accurate weight and computational load. Although the other proposed method, called the total difference-based fuzzy AHP (TDFAHP), gave better results than the frequently used Chang's fuzzy extent analysis method, it could not produce more accurate weight results than many other methods in general. But performance analysis shows that it is as good as the MFAHP in terms of computational load.     

A method of determining attribute weights in evidential reasoning approach based on incompatibility among attributes

Decisions on attribute weights are important problems in multiple attribute decision making. Many methods have been proposed to create attribute weights which are used to aggregate attributes in a simple additive weighting way. In this paper, a method of deriving attribute weights from incompatibility among attributes and possible constraints on the weights is developed based on the evidential reasoning approach in which attribute aggregation is nonlinear rather than linear. The incompatibility is a flexible combination of deviation incompatibility and decision incompatibility with a relaxation coefficient. The deviation incompatibility measures differences between assessments of alternatives on each attribute and the decision incompatibility quantifies differences between assessments of alternatives on one attribute and the aggregated assessments of the alternatives. For a specific alternative, two pairs of optimization problems with a constraint on the difference between potential weights and the combination of deviation incompatibility and decision incompatibility are designed to generate the favorable intervals of attribute weights and those of utilities of assessment grades. A problem of car performance assessment is investigated to demonstrate the applicability of the proposed method. The method is validated by comparison with other methods of producing attribute weights using the problem.     

On the priority vector associated with a reciprocal relation and a pairwise comparison matrix

We propose two straightforward methods for deriving the priority vector associated with a reciprocal relation, by some authors called fuzzy preference relation. Then, using transformations between pairwise comparison matrices and reciprocal relations, we study the relationships between the priority vectors associated with these two types of preference relations. Eventually, we show a brief example involving the newly introduced characterizations.     

An integrated logarithmic fuzzy preference programming based methodology for optimum maintenance strategies selection

Selecting optimum maintenance strategies plays a key role in saving cost, and improving the system reliability and availability. Analytic hierarchical process (AHP) is widely used for maintenance strategies selection in the Multiple Criteria Decision-Making (MCDM) field. But the traditional or hybrid AHP methods either produce multiple, even conflict priority results, or have complicated algorithm structures which are unstable to obtain the optimum solution. Therefore, this paper proposes an integrated Logarithmic Fuzzy Preference Programming (LFPP) based methodology in AHP to solve the optimum maintenance strategies selection problem. The multiplicative constraints and deviation variables are applied instead of additive ones to utilize both qualitative and quantitative data, and process the upper and lower triangular fuzzy judgments to obtain the same priorities. The proposed methodology can produce the unique normalized optimal priority vector for fuzzy pairwise comparison matrices, and it is capable of processing all comparison matrices to obtain the global priorities simultaneously and directly in the form of super-matrix according to the different requirements and judgments of decision-makers. Finally, an example is provided to demonstrate the feasibility and validity of the proposed methodology.     

Interval Evaluations in the Analytic Hierarchy Process By Possibility Analysis

Since a pairwise comparison matrix in the Analytic Hierarchy Process (AHP) is based on human intuition, the given matrix will always include inconsistent elements violating the transitivity property. We propose the Interval AHP by which interval weights can be obtained. The widths of the estimated interval weights represent inconsistency in judging data. Since interval weights can be obtained from inconsistent data, the proposed Interval AHP is more appropriate to human judgment. Assuming crisp values in a pairwise comparison matrix, the interval comparisons including the given crisp comparisons can be obtained by applying the Linear Programming (LP) approach. Using an interval preference relation, the Interval AHP for crisp data can be extended to an approach for interval data allowing to express the uncertainty of human judgment in pairwise comparisons.     

Data envelopment analysis for weight derivation and aggregation in the analytic hierarchy process

Data envelopment analysis (DEA) is proposed in this paper to generate local weights of alternatives from pair-wise comparison judgment matrices used in the analytic hierarchy process (AHP). The underlying assumption behind the approach is explained, and some salient features are explored. It is proved that DEA correctly estimates the true weights when applied to a consistent matrix formed using a known set of weights. DEA is further proposed to aggregate the local weights of alternatives in terms of different criteria to compute final weights. It is proved further that the proposed approach, called DEAHP in this paper, does not suffer from rank reversal when an irrelevant alternative(s) is added or removed.     

Combining different prioritization methods in the analytic hierarchy process synthesis

A multicriteria approach for combining prioritization methods within the analytic hierarchy process (AHP) is proposed. The leading assumption is that for each particular decision problem and related hierarchy, AHP must not necessarily employ only one prioritization method (e.g. eigenvector method). If more available methods are used to identify the best estimates of local priorities for each comparison matrix in the hierarchy, then the estimate of final alternatives’ priorities should also be the best possible, which is in natural concordance with an additive compensatory structure of the AHP synthesis. The most popular methods for deriving priorities from comparison matrices are identified as candidates (alternatives) to participate in AHP synthesis: additive normalization, eigenvector, weighted least-squares, logarithmic least-squares, logarithmic goal programming and fuzzy preference programming. Which method will be used depends on the result of multicriteria evaluation of their priority vectors’ performance with regard to suggested deviation and rank reversal measures. Two hierarchies with matrices of size 3–6 are used to illustrate an approach.     

A common framework for deriving preference values from pairwise comparison matrices

Pairwise comparison is commonly used to estimate preference values of finite alternatives with respect to a given criterion. We discuss 18 estimating methods for deriving preference values from pairwise judgment matrices under a common framework of effectiveness: distance minimization and correctness in error free cases. We point out the importance of commensurate scales when aggregating all the columns of a judgment matrix and the desirability of weighting the columns according to the preference values. The common framework is useful in differentiating the strength and weakness of the estimated methods. Some comparison results of these 18 methods on two sets of judgment matrices with small and large errors are presented. We also give insight regarding the underlying mathematical structure of some of the methods.Scope and purposePairwise comparison is commonly used to estimate preference values of finite alternatives with respect to a given criterion. This is part of the model structure of the analytical hierarchy process, a widely used multicriteria decision-making methodology. The main difficulty is to reconcile the inevitable inconsistency of the pairwise comparison matrix elicited from the decision makers in real-world applications. We discuss 18 estimating methods for deriving preference values from pairwise judgment matrices under a common framework of effectiveness: the common concepts of minimizing aggregated deviation and correctness in error free cases. The common framework is useful in differentiating the strength and weakness of these methods. For each of these methods, we point out their individual strength in decisional effectiveness. Some comparison results of these 18 methods on two sets of judgment matrices with small and large errors are presented. We also give insight regarding the underlying mathematical structure of some of the methods. We recommend the simple geometric mean method with the stronger feature of distance minimization and the simple normalized column sum method that is based on the simple ideas of commensurate unit and column sum. These two methods have closed-form formulas for easy calculation and good performance on both sets of judgment matrices with small and large errors.     

A GP-AHP method for solving group decision-making fuzzy AHP problems

The analytic hierarchy process (AHP) elicits a corresponding priority vector interpreting the preferred information from the decision-maker(s), based on the pairwise comparison values of a set of objects. Since pairwise comparison values are the judgments obtained from an appropriate semantic scale, in practice the decision-maker(s) usually give some or all pair-to-pair comparison values with an uncertainty degree rather than precise ratings. By employing the property of goal programming (GP) to treat a fuzzy AHP problem, this paper incorporates an absolute term linearization technique and a fuzzy rating expression into a GP-AHP model for solving group decision-making fuzzy AHP problems. In contrast to current fuzzy AHP methods, the GP-AHP method developed herein can concurrently tackle the pairwise comparison involving triangular, general concave and concave–convex mixed fuzzy estimates under a group decision-making environment.

Scope and purpose

Many real world decision problems involve multiple criteria in qualitative domains. As expected, such problems will be increasingly modeled as multiple criteria decision-making problems, which involve scoring on subjective/qualitative domains. This results in a class of significant problems for which an evaluation framework, which handles occurrences of seeming intransitivity and inconsistency will be required. Another interesting issue of group decision-making analysis is how to deal with disagreements between two or more different rankings within an alternative set. These phenomena are likely to appear in qualitative/subjective domains where the decision-making environment is ambiguous and vague. Therefore, this study proposes a GP-AHP model that is sufficiently robust to permit conflict and imprecision. Numerical examples demonstrate the effectiveness and applicability of the proposed models in deriving the most promising priority vector from a fuzzy AHP problem within a group decision-making environment.     

Ordinal Priority Approach (OPA) in Multiple Attribute Decision-Making

The current study aims to present a new method called Ordinal Priority Approach (OPA) in Multiple Attribute Decision-Making (MADM). This method can be used in individual or group decision-making (GDM). In the case of GDM, through this method, we first determine the experts and their priorities. The priority of experts may be determined based on their experience and/or knowledge. After prioritization of the experts, the attributes are prioritized by each expert. Meanwhile, each expert ranks the alternatives based on each attribute, and the sub-attributes if any. Ultimately, by solving the presented linear programming model of this method, the weights of the attributes, alternatives, experts, and sub-attributes would be obtained simultaneously. A significant advantage of the proposed method is that it does not make use of pairwise comparison matrix, decision-making matrix (no need for numerical input), normalization methods, averaging methods for aggregating the opinions of experts (in GDM) and linguistic variables. Another advantage of this method is the possibility for experts to only comment on the attributes and alternatives for which they have sufficient knowledge and experience. The validity of the proposed model has been evaluated using several group and individual instances. Finally, the proposed method has been compared with other methods such as AHP, BWM, TOPSIS, VIKOR, PROMETHEE and QUALIFLEX. Based on comparisons among the weights and ranks using Spearman and Pearson correlation coefficients, the proposed method has an applicable performance compared with other methods.     

Fuzzy analytic hierarchy process and analytic network process: An integrated fuzzy logarithmic preference programming

This paper proposes a two-stage fuzzy logarithmic preference programming with multi-criteria decision-making, in order to derive the priorities of comparison matrices in the analytic hierarchy pprocess (AHP) and the analytic network process (ANP). The Fuzzy Preference Programming (FPP) proposed by Mikhailov and Singh [L. Mikhailov, M.G. Singh, Fuzzy assessment of priorities with application to competitive bidding, Journal of Decision Systems 8 (1999) 11–28] is suitable for deriving weights in interval or fuzzy comparison matrices, especially those displaying inconsistencies. However, the weakness of the FPP is that it obtains priorities of comparison matrices by additive constraints, and generates different priorities by processing upper and lower triangular judgments. In addition, the FPP solves the comparison matrix individually. By using multiplicative constraints, the method proposed in this paper can generate the same priorities from upper and lower triangular judgments with crisp, interval or fuzzy values. Our proposed method can solve all of the matrices simultaneously by multiple objective programming. Finally, five examples are demonstrated to show the proposed method in more detail.     

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.