Paired Comparisons Of Alternatives Research Articles

Метод анализа иерархий: подход, основанный на теории латентных переменных

The paper proposes a mathematical model for evaluating the attractiveness of alternatives by a qualitative criterion, which is based on the method of evaluating latent variables according to the Rasch model. This model is intended for assessing the attractiveness of alternatives in expert evaluation using the hierarchy analysis method based on a paired comparison of alternatives. The proposed method will make it possible to obtain objective estimates of alternatives on a linear scale, which do not depend on the composition of the alternatives being evaluated. Aim. The purpose of the study is to describe a new mathematical apparatus that allows one to obtain estimates of the attractiveness of alternatives and has a number of advantages over traditional methods of evaluation. The proposed model is based on the theory of latent variables, and more precisely, on the Rasch model for estimating latent variables. Materials and methods. The basis of the model for evaluating alternatives described in the paper is the method of paired comparisons. However, unlike the traditional mathematical apparatus of the hierarchy analysis method, the method for identifying the preference of one alternative over another for each of their pairs is based on a probabilistic approach, so preference estimates are less abstract and more objective. The paper gives a mathematical substantiation of the proposed model, compares the results of estimation by the proposed model with the results obtained by traditional methods. Computational experiments are also described that substantiate the adequacy of the obtained estimates. Results. A mathematical model has been developed, which is an alternative to the hierarchy analysis method (T. Saaty) for assessing the attractiveness of alternatives using the method of paired comparisons. Unlike the traditional evaluation method, the model allows you to obtain independent estimates of alternatives on a linear scale, and the criteria for comparing alternatives are probabilistic in nature, which allows for an objective assessment. Computational experiments have shown the adequacy and good stability of the obtained estimates to changes in the initial data. Conclusion. The proposed method for processing expert information obtained from paired comparisons of alternatives will make it possible to obtain the most objective assessments of them and can be used as a mathematical apparatus for decision support systems in many areas of scientific and practical activity.

Multicriteria Decision Support System for Selecting the Structural and Functional Scheme of a Biogas Plant in the Agro-Industrial Complex

The variants of structural and functional schemes of biogas plants (BP) were investigated to obtain biogas and fertilizers based on the processing of solid organic waste in the agro-industrial complex (AIC). Based on the results of the study of biogas technologies, criteria are proposed for determining the best layout from the expert’s point of view. The systematization of options is given and alternatives are obtained. The method of multicriteria analysis of options based on fuzzy sets was used as a decision-making method. The use of the decision support system to select the best variant of the structural and functional scheme of the biogas plant predetermined the formation of a matrix of paired comparisons of dimension 5x5. To ensure high consistency of preferences associated with the procedure for filling in the matrices of paired comparisons of alternatives according to criteria, the method developed by us has been applied. This reduces the number of inquiries to the decision-maker. The selected structural and functional scheme meets the requirements for the development of typical automation solutions, which contributes to the replication of control systems for the same type of plants.

Two Hesitant Multiplicative Decision-Making Algorithms and Their Application to Fog-Haze Factor Assessment Problem

Hesitant multiplicative preference relation (HMPR) is a useful tool to cope with the problems in which the experts utilize Saaty’s 1–9 scale to express their preference information over paired comparisons of alternatives. It is known that the lack of acceptable consistency easily leads to inconsistent conclusions, therefore consistency improvement processes and deriving the reliable priority weight vector for alternatives are two significant and challenging issues for hesitant multiplicative information decision-making problems. In this paper, some new concepts are first introduced, including HMPR, consistent HMPR and the consistency index of HMPR. Then, based on the logarithmic least squares model and linear optimization model, two novel automatic iterative algorithms are proposed to enhance the consistency of HMPR and generate the priority weights of HMPR, which are proved to be convergent. In the end, the proposed algorithms are applied to the factors affecting selection of fog-haze weather. The comparative analysis shows that the decision-making process in our algorithms would be more straight-forward and efficient.

Features of the Regional Economic Systems Development with a Mono-Branch Focus

Subject . The relevance of the study is explained by the need to develop regional economic systems based on the activation of innovative processes and the use of a differentiated approach depending on the level of economic development of the territory. The subject of this study is the regions of the national economy, characterized by a strong branch orientation. Purpose . The author of the article attempts to develop an approach to ensure innovative development of economy on the basis of determining the strategic direction of regions development with an industry focus, by targeting the key parameters of the system. Within the framework of the stated goal solution, the method of analytical hierarchical procedure was used, allowing to decompose the investigated problem and to carry out a paired comparison of the selected development strategies. Methodology . A grouping of indicators was carried out, the dynamics of which can be used to determine the stage of the industry life cycle, dominating in the region. Three strategic vectors of the region development were identified, taking into account the dynamics of innovative processes in the industry and contribution to the region economic development which is dominant in the economic activity, the indicators determining the stage of the industry life cycle were pointed out, the approach to identify strategic development priorities of the regional economy was proposed taking into account the most important factors. There is an approach to the prioritization of regional economic systems development strategies with paired comparisons of alternatives. Results . The calculations allow choosing development strategies for the regional economy development with a mono-branch focus depending on the relative importance of the selected criteria. Conclusions . The formulated provisions can be taken into account in the development of the strategy of the regional economy advanced development by enhancing innovative activity.

Goal programming approach to derive intuitionistic multiplicative weights based on intuitionistic multiplicative preference relations

The intuitionistic multiplicative preference relation (IMPR) was introduced by Xia et al. [25] to deal with situations in which the decision makers (DMs) exhibit the characteristics of affirmation, negation and hesitation for the preference degrees over paired comparisons of alternatives. The IMPR can reflect the preference information of the DMs over alternatives more comprehensively than the multiplicative preference relation (MPR). In this paper, a new method for decision making is proposed to derive normalized intuitionistic multiplicative weights on the basis of the order consistent IMPR and the consistent IMPR. We first define the concepts of order consistent IMPR, consistent IMPR and normalized intuitionistic multiplicative weights, and then discuss some properties of the consistent IMPR. After that, we investigate a transformation formula to convert the normalized intuitionistic multiplicative priority weights into a consistent IMPR. An optimization model is constructed to generate the normalized intuitionistic multiplicative weights of IMPR, and the optimal deviation values obtained from the model enable us to improve the consistency of the given IMPR, such that the repaired IMPR is consistent. In the end, a numerical example is provided, and comparative analysis with Xu’s approach [26] is performed to demonstrate the validity and applicability of the proposed method.

Approaches to decision making with linguistic preference relations based on additive consistency

The linguistic preference relation (LPR) is introduced to efficiently deal with situations in which the decision makers (DMs) provide their preference information by using linguistic labels over paired comparisons of alternatives. However, the lack of consistency in decision making with LPRs can lead to inconsistent conclusions. In this paper, two new decision making methods are developed to improve the additive consistency of LPRs until they are acceptable, and eventually obtain the reliable decision making results. First, the new concepts of order consistency and additive consistency of LPRs are introduced, and followed by a discussion of the characterization about additive consistent LPRs. Then, a consistency index is defined to measure whether an LPR is of acceptable additive consistency. For an unacceptable additive consistent LPR, two automatic iterative algorithms are further proposed to help DMs improve additive consistency level until it is acceptable. In addition, the proposed algorithms can derive the priority weight vector from LPRs and obtain the ranking of the alternatives. Finally, the proposed methods are applied to an emergency operating center (EOC) selection problem. The comparative analysis demonstrates the applicability and effectiveness of the proposed methods.

Some Hesitant Multiplicative Aggregation Operators and Their Application in Group Decision Making with Hesitant Multiplicative Preference Relations

The hesitant multiplicative preference relation (HMPR) is a newly developed preference structure that uses an unsymmetrical scale (Saaty’s 1–9 scale) to express the decision makers’ (DMs’) preferences instead of the symmetrical scale (0.1–0.9 scale) in a hesitant fuzzy preference relation. This new preference relation is suitable for describing the situation in which the DMs are hesitant about several possible multiplicative values for the preference degrees over paired comparisons of alternatives. In this paper, we first define some new operational laws for hesitant multiplicative elements. Then, based on these operational laws, we develop two types of hesitant multiplicative aggregation operators for aggregating the hesitant multiplicative preference information, which we call the independent hesitant multiplicative aggregation operators and the correlative hesitant multiplicative aggregation operators. The independent hesitant multiplicative aggregation operators are based on the assumption that the aggregated arguments are independent, and the operators can be used to address situations where the preferences of the DMs are independent. The correlative hesitant multiplicative aggregation operators are developed to reflect the correlations of the aggregated arguments based on the Choquet integral and the power average, and the operators can be used to address the situation where the preferences of DMs are dependent on each other. We further give some desirable properties and special cases of the developed operators and apply these operators to develop several approaches to group decision making based on HMPRs. Finally, several practical examples are provided to illustrate the developed operators and approaches.

Approaches to group decision making with intuitionistic fuzzy preference relations based on multiplicative consistency

The intuitionistic fuzzy preference relation (IFPR) was introduced by Xu to efficiently deal with situations in which the decision makers (DMs) exhibit the characteristics of affirmation, negation and hesitation for the preference degrees over paired comparisons of alternatives. In this paper, two new approaches to group decision making (GDM) are proposed to derive the normalized intuitionistic fuzzy priority weights from IFPRs based on the order consistency and the multiplicative consistency. First, the concepts of order consistency and weak transitivity for IFPRs are introduced, and followed by a discussion of their desirable properties. Then, in order to convert the normalized intuitionistic fuzzy priority weights into multiplicative consistent IFPR, a transformation approach is investigated. Two linear optimization models are further developed to derive the normalized intuitionistic fuzzy weight vector for both individual and group IFPRs with the principle of minimizing the deviations between any provided IFPR and the converted multiplicative consistent IFPR, and the optimal deviation values obtained from the models enable us to improve the multiplicative consistency of IFPRs. Finally, based on the order consistency and the multiplicative consistency, two new algorithms for GDM are presented. Several numerical examples are provided, and comparative analyses with existing approaches are performed to demonstrate that the proposed methods are both valid and practical to deal with group decision making problems.

On the use of multiplicative consistency in hesitant fuzzy linguistic preference relations

As a new preference structure, the hesitant fuzzy linguistic preference relation (HFLPR) was recently introduced by Rodríguez, Martínez, and Herrera to efficiently address situations in which the decision makers (DMs) are hesitant about several possible linguistic terms for the preference degrees over paired comparisons of alternatives. In this paper, we define the multiplicative consistency of HFLPRs to ensure that the DMs are being neither random nor illogical, and propose a characterization about the multiplicative consistency of HFLPRs. A consistency index is defined to measure the deviation degree between a HFLPR and its multiplicative consistent HFLPR. For a HFLPR with unacceptably multiplicative consistency, we develop a consistency-improving process to adjust it into an acceptably multiplicative one. Moreover, we use the hesitant fuzzy linguistic aggregation operators to aggregate preferences in the acceptably multiplicative HFLPR to obtain the ranking results. Several illustrative examples are further provided to verify the developed methods. Finally, a comparison with other methods in the existing literature is performed to illustrate the advantages of the new methods.

Independent, convergent, and divergent decision behaviour for interactive multiple objectives linear programming

An interactive paired comparison approach for multiple objective linear programming (MOLP) problems is developed where the Decision Maker's preferential behaviour is presented by a utility function. The preferential behaviour can be independent, convergent, or divergent; which are mathematically presented by additive, quasi-concave, or quasi-convex utility functions respectively. The approach to find the best alternative consists of four phases: I. An additive utility function. II. A quasi-concave utility function. III. A quasi-convex utility function. IV. A feasible goal utility function. In the first three phases a paired comparison of alternatives are used and in Phase IV a feasible goal method is developed that finds the closest feasible efficient point for a given goal by the decision maker. This method also identifies resources required to make a given goal feasible. The approach can be extended to solve multiple objective integer/nonlinear optimization problems. Some experiments and examples are provided.

Hilbert's metric and the analytic hierarchy process

This paper explores some of the properties of Hilbert's projective metric as a measure of closeness between two ratio scales in the context of the Analytic Hierarchy Process. Smallperturbation arguments are used to contrast the sensitivity and the distributional behavior of this metric with the more traditional Euclidean distance function, in situations where the paired comparison of alternatives is subject to random perturbations, and priorities are estimated either by Saaty's eigenvalue method or by the logarithmic least squares principle. A pivotal property of Hilbert's metric has surfaced which allows for the construction of confidence regions for an underlying priority vector. These regions are seen to enjoy good coverage properties.

An interactive integrated multiobjective optimization approach for quasiconcave / quasiconvex utility functions

In this paper, a new interactive, integrated approach for solving multiobjective optimization problems is presented. The approach is general in that it handles two broad classes of implicit utility functions: quasiconcave and quasiconvex. The first step is to use preference comparison-based tests to determine the class of utility function that is consistent with the Decision Maker (DM's) underlying preferences with respect to a sample of nondominated alternatives. Then one of the imbedded algorithms that is appropriate for the selected utility function is used. In the case of quasiconcave utility, a modified Geoffrion-Dyer-Feinberg algorithm [7] is applied. It projects the gradient-based improvement direction on the nondominated frontier and provides an interactive termination criterion. The quasiconvex utility-based algorithms chosen depends on the structure of the feasible set. The demands upon the DM are kept to a minimum in the sense that only paired comparisons of alternatives and trade-off evaluations are elicited by all the algorithms. An example is presented for the quasiconcave utility-based algorithm.

An Interactive Multiple Criteria Approach for Parameter Selection in Metal Cutting

Many real-world problems, particularly in manufacturing planning, can be expressed as multiple criteria problems. In this paper, we formulate a metal cutting operation, the turning operation, as a discrete multiple objective problem. The objectives are to minimize cost per part, production time per part, and roughness of the work surface, simultaneously. We discuss a heuristic, gradient-based multiple criteria decision making approach that we apply to parameter selection in metal cutting. The approach can be applied to any multicriteria problems with either discrete, integer, or continuous sets of alternatives. The approach is interactive and requires only paired comparisons of alternatives. For the metal cutting problem, we show how efficient alternatives can be generated by a discrete variable approach and how the gradient-based multiple objective approach can be implemented to obtain the most preferred alternative. We also discuss the software package for microcomputers as a decision support system for parameter selection. The newly developed Computer Aided Machine Parameter Selection (CAMPS) package is then compared to some of the recent commercial packages on the market. Some computational experiments are reported and some examples are discussed.

A decision support system and a heuristic interactive approach for solving discrete multiple criteria problems

The paper consists of two parts: (1) a decision-support system (DSS) for discrete multiple criteria decision-making (MCDM) problems; and (2) a heuristic interactive approach when the decision-maker (DM) responds to paired comparisons of alternatives. In the DSS, the DM is provided with flexibility in terms of using different approaches, different assumptions, and manipulating and controlling the solution procedure in a very simple and convenient way. In the interactive approach it is assumed that the DM wishes to maximize an unknown quasiconcave utility function for discrete MCDM problems. Several heuristic procedures are developed for ranking and eliminating alternatives, assuming that the DM can make paired comparisons and indicate strength of preference. An interactive paired comparison method is developed using these heuristics and a gradient-based procedure. Heuristic approaches for finding local adjacent discrete points to approximate the gradient and for identifying discrete points for the one-dimensional search are discussed. >

Group Decision Problem Under Competitive Situation and its Interactive Solution by Optimization-Satisfaction Approach

This paper concerns a group decision problem for competitive system where plural decision-makers exist. The group must choose the alternative that is “best for the group” in some sense. The problem is formulated as follows. Each decision-maker is guaranteed with the satisfaction condition such that his objective function should be always kept below a certain permissible level, regardless of whatever opponents’ decisions may be taken. All decision-makers then cooperate together and seek a group preference solution to minimize a group’s objective function under the satisfaction condition. Hence a two-level optimization problem whose constraints include maximizations by the opponents is presented.Our primary aim is to develop a method for solving the described problem by means of usual non-linear programming. The second is to find methods to aggregate group preference. The group preference solution is obtained by using paired comparisons of alternatives and the Borda's rating method. Utilizing mathematical programming techniques and a computer-assisted system, the group decision problem under competitive situation can be solved in man-machine interaction scheme.