Combining compound linguistic ordinal scale and cognitive pairwise comparison in the rectified fuzzy TOPSIS method for group decision making

Group decision making is the process to explore the best choice among the screened alternatives under predefined criteria with corresponding weights from assessment of a group of decision makers. The Fuzzy TOPSIS taking an evaluated fuzzy decision matrix as input is a popular tool to analyze the ideal alternative. This research, however, finds that the classical fuzzy TOPSIS produces a misleading result due to some inappropriate definitions, and proposes the rectified fuzzy TOPSIS addressing two technical problems. As the decision accuracy also depends on the evaluation quality of the fuzzy decision matrix comprising rating scores and weights, this research applies compound linguistic ordinal scale as the fuzzy rating scale for expert judgments, and cognitive pairwise comparison for determining the fuzzy weights. The numerical case of a robot selection problem demonstrates the hybrid approach leading to the much reliable result for decision making, comparing with the conventional fuzzy Analytic Hierarchy Process and TOPSIS.     

A fuzzy ranking method with range reduction techniques

For ranking alternatives based on pairwise comparisons, current analytic hierarchy process (AHP) methods are difficult to use to generate useful information to assist decision makers in specifying their preferences. This study proposes a novel method incorporating fuzzy preferences and range reduction techniques. Modified from the concept of data envelopment analysis (DEA), the proposed approach is not only capable of treating incomplete preference matrices but also provides reasonable ranges to help decision makers to rank decision alternatives confidently.     

Fuzzy analytic hierarchy process: A logarithmic fuzzy preference programming methodology

Fuzzy analytic hierarchy process (AHP) proves to be a very useful methodology for multiple criteria decision-making in fuzzy environments, which has found substantial applications in recent years. The vast majority of the applications use a crisp point estimate method such as the extent analysis or the fuzzy preference programming (FPP) based nonlinear method for fuzzy AHP priority derivation. The extent analysis has been revealed to be invalid and the weights derived by this method do not represent the relative importance of decision criteria or alternatives. The FPP-based nonlinear priority method also turns out to be subject to significant drawbacks, one of which is that it may produce multiple, even conflict priority vectors for a fuzzy pairwise comparison matrix, leading to entirely different conclusions. To address these drawbacks and provide a valid yet practical priority method for fuzzy AHP, this paper proposes a logarithmic fuzzy preference programming (LFPP) based methodology for fuzzy AHP priority derivation, which formulates the priorities of a fuzzy pairwise comparison matrix as a logarithmic nonlinear programming and derives crisp priorities from fuzzy pairwise comparison matrices. Numerical examples are tested to show the advantages of the proposed methodology and its potential applications in fuzzy AHP decision-making.     

A fuzzy extension of Analytic Hierarchy Process based on the constrained fuzzy arithmetic

The aim of the paper is to highlight the necessity of applying the concept of constrained fuzzy arithmetic instead of the concept of standard fuzzy arithmetic in a fuzzy extension of Analytic Hierarchy Process (AHP). Emphasis is put on preserving the reciprocity of pairwise comparisons during the computations. For deriving fuzzy weights from a fuzzy pairwise comparison matrix, we consider a fuzzy extension of the geometric mean method and simplify the formulas proposed by Enea and Piazza (Fuzzy Optim Decis Mak 3:39–62, 2004). As for the computation of the overall fuzzy weights of alternatives, we reveal the inappropriateness of applying the concept of standard fuzzy arithmetic and propose the proper formulas where the interactions among the fuzzy weights are taken into account. The advantage of our approach is elimination of the false increase of uncertainty of the overall fuzzy weights. Finally, we advocate the validity of the proposed fuzzy extension of AHP; we show by an illustrative example that by neglecting the information about uncertainty of intensity of preferences we lose an important part of knowledge about the decision making problem which can cause the change in ordering of alternatives.     

Modeling subjective evaluation for fuzzy group multicriteria decision making

This paper presents a new fuzzy multicriteria decision making (MCDM) approach for evaluating decision alternatives involving subjective judgements made by a group of decision makers. A pairwise comparison process is used to help individual decision makers make comparative judgements, and a linguistic rating method is used for making absolute judgements. A hierarchical weighting method is developed to assess the weights of a large number of evaluation criteria by pairwise comparisons. To reflect the inherent imprecision of subjective judgements, individual assessments are aggregated as a group assessment using triangular fuzzy numbers. To obtain a cardinal preference value for each decision alternative, a new fuzzy MCDM algorithm is developed by extending the concept of the degree of optimality to incorporate criteria weights in the distance measurement. An empirical study of aircraft selection is presented to illustrate the effectiveness of the approach.     

A new design of the elimination and choice translating reality method for multi-criteria group decision-making in an intuitionistic fuzzy environment

The purpose of this paper is to design a new extension of the ELECTRE, known as the elimination and choice translating reality method, for multi-criteria group decision-making problems based on intuitionistic fuzzy sets. This method is widely utilized when a set of alternatives should be identified and evaluated with respect to a set of conflicting criteria by reflecting decision makers’ (DMs’) preferences. However, handling the exact data and numerical measure is difficult to be precisely focused because the DMs’ judgments are often vague in real-life decision problems and applications. A more realistic and practical approach can be to use linguistic variables expressed in intuitionistic fuzzy numbers instead of numerical data to model DMs’ judgments and to describe the inputs in the ELECTRE method. The proposed intuitionsitic fuzzy ELECTRE utilizes the truth-membership function and non-truth-membership function to indicate the degrees of satisfiability and non-satisfiability of each alternative with respect to each criterion and the relative importance of each criterion, respectively. Then, a new discordance intuitionistic index is introduced, which is extended from the concept of the fuzzy distance measure. Outranking relations are defined by pairwise comparisons and a decision graph is depicted to determine which alternative is preferable, incomparable or indifferent in the intuitionistic fuzzy environment. Finally, a comprehensive sensitivity analysis is employed to further study regarding the impact of threshold values on the final evaluation, and a comparative analysis is demonstrated with an application example in flexible manufacturing systems between the proposed ELECTRE method and the existing intuitionistic fuzzy technique for order preference by similarity to ideal solution (IF-TOPSIS) method.     

Probabilistic judgments specified partially in the analytic hierarchy process

The Analytic Hierarchy Process (AHP) is a decision-making tool which yields priorities for decision alternatives. This paper proposes a new approach to elicit and synthesize expert assessments for the group decision process in the AHP. These new elicitations are given as partial probabilistic specifications of the entries of pairwise comparisons matrices. For a particular entry of the matrix, the partial probabilistic elicitations could arise in the form of either probability assignments regarding the chance of that entry falling in specified intervals or selected quantiles for that entry. A new class of models is introduced to provide methods for processing this partial probabilistic information. One advantage of this approach is that it allows to generate as many pairwise comparison matrices of the decision alternatives as one desires. This, in turn, allows us to determine the statistical significance of the priorities of decision alternatives.     

Extensions of the multicriteria analysis with pairwise comparison under a fuzzy environment

Multicriteria decision-making (MCDM) problems often involve a complex decision process in which multiple requirements and fuzzy conditions have to be taken into consideration simultaneously. The existing approaches for solving this problem in a fuzzy environment are complex. Combining the concepts of grey relation and pairwise comparison, a new fuzzy MCDM method is proposed. First, the fuzzy analytic hierarchy process (AHP) is used to construct fuzzy weights of all criteria. Then, linguistic terms characterized by L–R triangular fuzzy numbers are used to denote the evaluation values of all alternatives versus subjective and objective criteria. Finally, the aggregation fuzzy assessments of different alternatives are ranked to determine the best selection. Furthermore, this paper uses a numerical example of location selection to demonstrate the applicability of the proposed method. The study results show that this method is an effective means for tackling MCDM problems in a fuzzy environment.     

A comparative study of the numerical scales and the prioritization methods in AHP

In the analytic hierarchy process (AHP), a decision maker first gives linguistic pairwise comparisons, then obtains numerical pairwise comparisons by selecting certain numerical scale to quantify them, and finally derives a priority vector from the numerical pairwise comparisons. In particular, the validity of this decision-making tool relies on the choice of numerical scale and the design of prioritization method. By introducing a set of concepts regarding the linguistic variables and linguistic pairwise comparison matrices (LPCMs), and by defining the deviation measures of LPCMs, we present two performance measure algorithms to evaluate the numerical scales and the prioritization methods. Using these performance measure algorithms, we compare the most common numerical scales (the Saaty scale, the geometrical scale, the Ma–Zheng scale and the Salo–Hämäläinen scale) and the prioritization methods (the eigenvalue method and the logarithmic least squares method). In addition, we also discuss the parameter of the geometrical scale, develop a new prioritization method, and construct an optimization model to select the appropriate numerical scales for the AHP decision makers. The findings in this paper can help the AHP decision makers select suitable numerical scales and prioritization methods.     

Type-2 fuzzy numbers made simple in decision making

Fuzzy Optimization and Decision Making - For the decision-making problems based on decision makers’ judgments in terms of linguistic terms, we propose type-2 fuzzy numbers (T2FNs) that allow...     

Estimation of missing judgments in AHP pairwise matrices using a neural network-based model

Selecting relevant features to make a decision and expressing the relationships between these features is not a simple task. The decision maker must precisely define the alternatives and criteria which are more important for the decision making process. The Analytic Hierarchy Process (AHP) uses hierarchical structures to facilitate this process. The comparison is realized using pairwise matrices, which are filled in according to the decision maker judgments. Subsequently, matrix consistency is tested and priorities are obtained by calculating the matrix principal eigenvector. Given an incomplete pairwise matrix, two procedures must be performed: first, it must be completed with suitable values for the missing entries and, second, the matrix must be improved until a satisfactory level of consistency is reached. Several methods are used to fill in missing entries for incomplete pairwise matrices with correct comparison values. Additionally, once pairwise matrices are complete and if comparison judgments between pairs are not consistent, some methods must be used to improve the matrix consistency and, therefore, to obtain coherent results. In this paper a model based on the Multi-Layer Perceptron (MLP) neural network is presented. Given an AHP pairwise matrix, this model is capable of completing missing values and improving the matrix consistency at the same time.     

The Analytic Hierarchy Process in an uncertain environment: A simulation approach

Traditionally, decision makers were forced to converge ambiguous judgments to a single point estimate in order to describe a pairwise relationship between alternatives relative to some criterion for use in the Analytic Hierarchy Process (AHP). Since many circumstances exist which make such a convergence difficult, confidence in the outcome of an ensuing AHP synthesis may be reduced. Likewise, when a group of decision makers cannot arrive at a consensus regarding a judgment, some members of the group may simply lose confidence in the overall synthesis if they lack faith in some of the judgments. The AHP utilizes point estimates in order to derive the relative weights of criteria, sub-criteria, and alternatives which govern a decision problem. However, when point estimates are difficult to determine, distributions describing feasible judgments may be more appropriate. Using simulation, we will demonstrate that levels of confidence can be developed, expected weights can be calculated and expected ranks can be determined. It will also be shown that the simulation approach is far more revealing than traditional sensitivity analysis.     

Strong reciprocity and strong consistency in pairwise comparison matrix with fuzzy elements

The decision making problem considered in this paper is to rank n alternatives from the best to the worst, using the information given by the decision maker in the form of an \(n\times n\) pairwise comparison matrix. Here, we deal with pairwise comparison matrices with fuzzy elements. Fuzzy elements of the pairwise comparison matrix are applied whenever the decision maker is not sure about the value of his/her evaluation of the relative importance of elements in question. We investigate pairwise comparison matrices with elements from abelian linearly ordered group (alo-group) over a real interval. The concept of reciprocity and consistency of pairwise comparison matrices with fuzzy elements have been already studied in the literature. Here, we define stronger concepts, namely the strong reciprocity and strong consistency of pairwise comparison matrices with fuzzy intervals as the matrix elements (PCF matrices), derive the necessary and sufficient conditions for strong reciprocity and strong consistency and investigate their properties as well as some consequences to the problem of ranking the alternatives.     

Fuzzified AHP in the evaluation of scientific monographs

Fuzzification of the analytic hierarchy process (AHP) is of great interest to researchers since it is a frequently used method for coping with complex decision making problems. There have been many attempts to fuzzify the AHP. We focus particularly on the construction of fuzzy pairwise comparison matrices and on obtaining fuzzy weights of objects from them subsequently. We review the fuzzification of the geometric mean method for obtaining fuzzy weights of objects from fuzzy pairwise comparison matrices. We illustrate here the usefulness of the fuzzified AHP on a real-life problem of the evaluation of quality of scientific monographs in university environment. The benefits of the presented evaluation methodology and its suitability for quality assessment of R&D results in general are discussed. When the task of quality assessment in R&D is considered, an important role is played by peer-review evaluation. Evaluations provided by experts in the peer-review process have a high level of subjectivity and can be expected in a linguistic form. New decision-support methods (or adaptations of classic methods) well suited to deal with such inputs, to capture the consistency of experts’ preferences and to restrict the subjectivity to an acceptable level are necessary. A new consistency condition is therefore defined here to be used for expertly defined fuzzy pairwise comparison matrices.     

Fuzzy Application to the Analytic Hierarchy Process for Robot Selection

The problem of Robot Selection is of great relevance in the present times of automation. Traditionally such problems were addressed using conventional techniques of Multi Criteria Decision Making such as The Analytic Hierarchy Process (AHP) and The Multi Attribute Utility Theory (MAUT). This paper proposes a methodology for solving common Robot Selection problems using a modification of the conventional AHP by incorporating ‘Fuzzy Linguistic Variables’ in place of numbers. The methodology encapsulates creation of Fuzzy Interface for conversion of input and output variables into suitable linguistic variables. Further, employing the fuzzification process by assigning the linguistic variables to numerical values of the membership functions and formulating suitable decision rules, the procedure culminates into the defuzzification process for converting fuzzy output into crisp value and obtaining the result in the form of Fuzzy Score. The proposed model is explained using a numerical example. The paper also presents a validation of the proposed methodology over real world problems and provides directions for future research towards the end.Additional support was provided by the National Science Foundation.     

2-order additive fuzzy measure identification method based on diamond pairwise comparison and maximum entropy principle

Fuzzy measures can flexibly describe the relative importance of decision criterion as well as their interactions in multicriteria decision making. Based on the diamond pairwise comparison, a new identification method of 2-order additive fuzzy measure is proposed. The relative weight and the interaction degree can be obtained simultaneously for every pair of criteria in the diamond pairwise comparison. The Choquet integral-based equivalent alternative curve can help the decision maker estimate the interaction degrees between criteria. The overall importance of each criterion is obtained by the maximum eigenvector method of AHP. According to the maximum fuzzy measure entropy principal, a nonlinear programming is constructed to identify the interaction indices among criteria. Finally, an illustrative example shows the feasibility and validity of the proposed identification method.     

Fuzzy logic modifications of the Analytic Hierarchy Process

The Analytic Hierarchy Process (AHP) is a measurement methodology based on pair-wise comparisons that relies on judgment to derive priority scales. During its implementation, one constructs hierarchies, then makes judgments or performs measurements on pairs of elements with respect to a criterion to derive preference scales, which are then synthesized throughout the structure to select the preferred alternative.One of the areas where the AHP finds application is in the subjective phases of risk assessment (RA), where it is used to structure and prioritize diverse risk factors, including the judgments of experts. Since fuzzy logic (FL) has been shown to be an effective tool for accommodating human experts and their communication of linguistic variables, there has been research aimed at modeling the fuzziness in the AHP (FAHP), and recently the focus of some of that modeling has been with respect to RA.The literature discusses more than one FAHP model, which raises the question as to which are the prominent models and what are their characteristics. In response to this question, we examine three of the most influential FAHP models. The article proceeds as follows. It begins with a brief overview of the AHP and its limitations when confronted with a fuzzy environment. This is followed by a discussion of FL modifications of the AHP. A RA-based likelihood score example is used throughout. The article ends with a commentary on the findings.     

Ordinal judgments in multiattribute decision analysis

The article discusses the contradiction between the ambiguity of human judgment in a multicriterion environment and the exactness of the assessments required in the majority of the decision-making methods. Preferential information from the decision makers in the ordinal form (e.g., “more preferable”, “less preferable”, etc.) is argued to be more stable and more reliable than cardinal input. Ways of obtaining and using ordinal judgments for rank ordering of multiattribute alternatives are discussed. The effectiveness of the step-wise procedure of using ordinal tradeoffs for comparison of alternatives is evaluated. We introduce the notion of ordinal tradeoffs, presentation of ordinal tradeoffs as a flexible three-stage process, a paired joint ordinal scale (PJOS), and evaluation of the effectiveness of the three-stage process. Simulation results examine the sensitivity of the number of pairwise comparisons required for given numbers of criteria and categories within criteria, as well as the number of alternatives analyzed. This simulation shows that ordinal pairwise comparisons provide sufficient power to discriminate between 75% and 80% of the alternatives compared. While the proportional number of pairwise comparisons relative to the maximum possible decreases with the number of criteria and categories, the method is relatively insensitive to the number of alternatives considered.