Row Geometric Mean Research Articles

Right-left asymmetry of the eigenvector method: A simulation study

The eigenvalue method, suggested by the developer of the extensively used Analytic Hierarchy Process methodology, exhibits right-left asymmetry: the priorities derived from the right eigenvector do not necessarily coincide with the priorities derived from the reciprocal left eigenvector. This paper offers a comprehensive numerical experiment to compare the two eigenvector-based weighting procedures and their reasonable alternative of the row geometric mean with respect to four measures. The underlying pairwise comparison matrices are constructed randomly with different dimensions and levels of inconsistency. The disagreement between the two eigenvectors turns out to be not always a monotonic function of these important characteristics of the matrix. The ranking contradictions can affect alternatives with relatively distant priorities. The row geometric mean is found to be almost at the midpoint between the right and inverse left eigenvectors, making it a straightforward compromise between them.

Reducing incompatibility in a local AHP-group decision making context

In the context of local analytic hierarchy process-group decision making (AHP-GDM), this paper presents a theoretical framework and a semi-automatic procedure for reducing incompatibility between the actors involved in the decision making process and the collective position. The row geometric mean is employed as the prioritisation procedure and the geometric compatibility index (textrm{GCOMPI}) as the incompatibility measure; individual pairwise comparison matrices are considered as the input of the reduction process, whilst the collective vector is the output. The reduction is attained by slightly modifying, in relative terms, the judgements of the collective pairwise comparison matrix, irrespective of the method used to obtain it, that further improve the textrm{GCOMPI}. The resulting judgements of the collective matrix and the associated collective priorities are close to the initial collective values. The procedure does not modify the judgements of the initial individual matrices and this simplifies the process of reaching consensus. A simulation analysis is utilised to study the performance of the algorithm along with an illustrative numerical example. The analysis proves that the proposed algorithm is easy to implement and efficient, it provides mathematically closed results and significantly reduces the textrm{GCOMPI} associated with the precise consistency consensus matrix which is one of the AHP-GDM tools. The framework allows the procedure to be adapted to specific interests.

A Decision Support System for Improving the Inconsistency in AHP

This paper presents a DSS aimed at helping decision makers reduce and improve their inconsistency in eliciting their judgements when using the analytic hierarchy process (AHP). The DSS is designed for revising the judgements of a pairwise comparison matrix when the row geometric mean (RGM) is used as the prioritisation procedure and the geometric consistency index (GCI) as the inconsistency measure. The procedure employed guarantees that both the judgements and the derived priority vector will be close to the initial values. The DSS allows different degrees of participation of the decision maker in the review/modification of the judgements: no participation (automatic mode); prior participation (semi-automatic mode); and ongoing participation (interactive mode). The DSS also includes options to incorporate other requirements of the decision maker, such as limiting the modified values to an interval or improving inconsistency by modifying the lowest number of judgements, among others.

A new framework for the sustainable development goals of Saudi Arabia

Achieving sustainable development goals agenda 2030 is the aspiration of all the United Nation’s member countries. Countries have an uneven distribution of natural resources, economic strength, and capacity building. Many studies analyze these goals using various models and considering different aspects of the kingdom of Saudi Arabia's (KSA) economy. However, none of the studies modelled the key indicators mathematically to quantify the achievement level towards vision 2030; hence aims to bridge the existing literature gap In this paper, the sustainable development goals (SDGs) of KSA are considered, a mathematical model is formulated in light of fuzzy and weighted goal programming using membership function. The model comprises three goals related to the gross domestic product, sustainable energy consumption and employment capacity of the economy's contributing sectors. The analytic hierarchy process integrated to compute the goals’ weights using row geometric mean method. The study established the goals's satisfaction level with 57% overall achievement of the vision 2030. Individually, economic growth goal is 45% achievable. The clean energy consumption-related goal is 67% realizable. The employment-related goal is 78% attainable within the time frame of the vision 2030. The study suggests that KSA diversify its energy sector by concentrating and investing more resources in alternative energy sources, including renewable energy such as solar, wind, biomass, and nuclear energy. Also, there is a need to involve more vibrant and talented youths in the critical decision-making process. The model is simple and can be replicable in a similar country with slight modifications.

Revenue allocation in Formula One: a pairwise comparison approach

ABSTRACT A model is proposed to allocate Formula One World Championship prize money among the constructors. The methodology is based on pairwise comparison matrices, allows for the use of any weighting method, and makes possible to tune the level of inequality. We introduce an axiom called scale invariance, which requires the ranking of the teams to be independent of the parameter controlling inequality. The eigenvector method is revealed to violate this condition in our dataset, while the row geometric mean method always satisfies it. The revenue allocation is not influenced by the arbitrary valuation given to the race prizes in the official points scoring system of Formula One and takes the intensity of pairwise preferences into account, contrary to the standard Condorcet method. Our approach can be used to share revenues among groups when group members are ranked several times.

Balanced Scorecard Adaptation using AHP for the Corporate Healthcare Sector in India

The Balanced Scorecard(BSC) developed in the 1990s as a performance management tool finds significant applications in several areas including healthcare. Healthcare Industry though was slow in utilising the BSC due to complications in data sourcing and its management. The varied nature of data generated and handled in the process of healthcare delivery poses a challenge in measuring and prioritising the Key Performance Indicators(KPI). In the present study, to tackle this challenge the Analytic Hierarchy Process(AHP) which uses mathematical modelling and powerful synthesis for combining data and judgement to effectively rank options and predict outcomes has been applied. Here, the AHP was used to develop a comprehensive BSC that will not only give priorities within each perspective of the BSC but also provide an overall ranking list to aid performance management. Twenty senior administrators from a Corporate Hospital in India took part in the AHP exercise to give their pairwise choices for all the KPIs. The eigenvector method of the AHP aided in consolidating and processing their collective opinions to give a scientifically designed BSC framework. To ensure data validity in terms of consistencies and to perform the complex calculations involved, the Business Performance Management Singapore (BPMSG) template was used. By adopting the Row Geometric Mean Method this template provided the final output in an easy-to-read graphical form which can be displayed on a digital dashboard. The primary advantage of this method of display is the flexibility it permits in modifying the target values of the KPIs, real-time. The outcome of this study is expected to provide strategic directions to the hospital. The developed BSC framework is also expected to provide a benchmark for other similar healthcare institutions across India.

Reducing inconsistency measured by the geometric consistency index in the analytic hierarchy process

This paper presents a theoretical framework and a procedure for revising the judgements and improving the inconsistency of an Analytic Hierarchy Process (AHP) pairwise comparison matrix when the Row Geometric Mean (RGM) is used as the prioritisation procedure and the Geometric Consistency Index (GCI) is the inconsistency measure. Inconsistency is improved by slightly modifying the judgements that further reduce the GCI. Both the judgements and the derived priority vector will be close to the initial values. A simulation study is utilised to analyse the performance of the algorithm. The proposed framework allows the specification of the procedure to particular interests. A numerical example illustrates the proposed procedure.

Підходи до оцінки ефективності експертних методів

It is shown that definition of accuracy of expert methods represents a significant problem due to unavailability of benchmark values of object estimates in real expert examinations. It is suggested to use the efficiency indicator based on the stability of results of aggregation methods under deviations of input data. It has been studied two approaches to expert method efficiency evaluation on the example of comparison of two modifications of combinatorial aggregation method (in one modification spanning tree weights are taken into consideration, while in the other they are not). The first approach is based on analysis of real data of expert estimation of special benchmark objects, while the second one is based on simulation of the whole expert examination cycle, including the estimates themselves. In the process of efficiency evaluation of the two specified methods, the simulation-based approach turned out to be more suitable and representative. The obtained experimental results empirically prove the advantage of combinatorial method, taking spanning tree weights into account, over the method where these weights are not considered (and, consequently, over row geometric mean and logarithmic least squares methods). They also allow us to draw several conceptual conclusions, such as:1) Accuracy of expert estimates and efficiency of expert data aggregation methods are not the same concept. Often low accuracy of expert data aggregation methods results from mistakes made by experts during estimation and not from the drawbacks of the method itself;2) Not only accuracy, but also consistency of expert estimates is important, both within the pair-wise comparison matrix provided by a single expert, and among pair-wise comparisons provided by an expert group; the more consistent the data is, the more credibility it deserves;3) Only conceptually different methods, using different input data, can be compared based on real expert estimates. If the input data is the same and only the methods of their aggregation are different, then in order to evaluate such methods, one should simulate the whole expert examination cycle.Tabl.: 5. Fig.: 6. Refs: 24 titles.

The (logarithmic) least squares optimality of the arithmetic (geometric) mean of weight vectors calculated from all spanning trees for incomplete additive (multiplicative) pairwise comparison matrices

ABSTRACTComplete and incomplete additive/multiplicative pairwise comparison matrices are applied in preference modelling, multi-attribute decision making and ranking. The equivalence of two well known methods is proved in this paper. The arithmetic (geometric) mean of weight vectors, calculated from all spanning trees, is proved to be optimal to the (logarithmic) least squares problem, not only for complete, as it was recently shown in Lundy, M., Siraj, S., Greco, S. (2017): The mathematical equivalence of the “spanning tree” and row geometric mean preference vectors and its implications for preference analysis, European Journal of Operational Research 257(1) 197–208, but for incomplete matrices as well. Unlike the complete case, where an explicit formula, namely the row arithmetic/geometric mean of matrix elements, exists for the (logarithmic) least squares problem, the incomplete case requires a completely different and new proof. Finally, Kirchhoff's laws for the calculation of potentials in electric circuits is connected to our results.

A characterization of the Logarithmic Least Squares Method

We provide an axiomatic characterization of the Logarithmic Least Squares Method (sometimes called row geometric mean), used for deriving a preference vector from a pairwise comparison matrix. This procedure is shown to be the only one satisfying two properties, correctness in the consistent case, which requires the reproduction of the inducing vector for any consistent matrix, and invariance to a specific transformation on a triad, that is, the weight vector is not influenced by an arbitrary multiplication of matrix elements along a 3-cycle by a positive scalar.

A Reliable Method for Consistency Improving of Interval Multiplicative Preference Relations Expressed under Uncertainty

When a diverse group of experts share their knowledge to ascertain and solve a particular multiple criteria decision-making (MCDM) problem, uncertainty arises from several sources. In those cases, the interval multiplicative preference relation (IMPR) approach is a useful technique when verifying consistency. As illustrated in this paper, the validation and the improvement of consistency require robust analysis tools and algorithms. The proposed methodology provides reliable and consistent IMPR, which can be quantified in terms of row geometric mean method (RGMM) or the eigenvalue method (EM). In this manner, decision makers’ choices are implicitly including their uncertainty while maintaining acceptable consistency. The present approach is based on the Hadamard’s dissimilarity operator and through an algorithm, the derivation of a reliable and consistent IMPR is synthesized. In order to illustrate our results and compare them appropriately with other methodologies, a few examples are addressed and solved.

Characterization of the Row Geometric Mean Ranking with a Group Consensus Axiom

An axiomatic approach is applied to the problem of extracting a ranking of the alternatives from a pairwise comparison ratio matrix. The ordering induced by row geometric mean method is proved to be uniquely determined by three independent axioms, anonymity (independence of the labelling of alternatives), responsiveness (a kind of monotonicity property) and aggregation invariance, which requires the preservation of group consensus, that is, the pairwise ranking between two alternatives should remain unchanged if unanimous individual preferences are combined by geometric mean.

The mathematical equivalence of the “spanning tree” and row geometric mean preference vectors and its implications for preference analysis

Pairwise comparison is a widely used approach to elicit comparative judgements from a decision maker (DM), and there are a number of methods that can be used to then subsequently derive a consistent preference vector from the DM’s judgements. While the most widely used method is the eigenvector method, the row geometric mean approach has gained popularity due to its mathematical properties and its ease of implementation. In this paper, we discuss a spanning tree method and prove the mathematical equivalence of its preference vector to that of the row geometric mean approach. This is an important finding due to the fact that it identifies an approach for generating a preference vector which has the mathematical properties of the row geometric mean preference vector, and yet, in its entirety, the spanning tree method has more to offer than the row geometric mean method, in that, it is inherently applicable to incomplete sets of pairwise comparison judgements, and also facilitates the use of statistical and visual techniques to gain insights into inconsistency in the DM’s judgements.

HodgeRank Is the Limit of Perron Rank

In the context of pairwise comparison ranking, we show that HodgeRank (row geometric mean), is the limit of Perron Rank (ranking with principal eigenvector) as a certain parameter k goes to 0. This result provides a novel mathematical link between two important pairwise ranking methods. It complements the known result that as k approaches infinity, Perron Rank converges to Tropical Rank. Thus, these three pairwise ranking methods belong to the same parametrized family. Our proof technique is useful for mathematical comparison of these methods. As a sample application, we show that for ranking models with i.i.d noise, HodgeRank is a linear approximation of Perron Rank. In this particular setup, for large numbers of items with sufficiently large score differences, the two methods yield identical ordinal rankings.

Some extensions of the precise consistency consensus matrix

The Precise Consensus Consistency Matrix (PCCM) is an AHP-Group Decision Making (AHP-GDM) tool, defined by Aguarón et al. [2] and developed in a local context (a single criterion) in which the decision makers are assigned the same weights. Using the Row Geometric Mean as the prioritisation procedure, consensus is sought between the different decision makers when the modifications of their initial positions or judgements are guaranteed to be within the range of values accepted for a given inconsistency level. This paper upgrades the algorithm initially proposed for obtaining the PCCM in two ways: (i) it considers the case of different weights for the decision makers; and (ii) it strengthens the idea of consistency in the design of the algorithm. One of the drawbacks of this decisional tool is that it is sometimes impossible to achieve a complete matrix. To address this, we propose a procedure for attaining a complete common consensus judgement matrix or, at least, a matrix with the minimum number of entries that are required to derive the priorities. Finally, we compare the results obtained when applying the extensions of the PCCM with those obtained using the two traditional procedures (AIJ and AIP) usually employed in AHP-GDM. In order to do this, we use a set of indicators that measure the violations in consistency of the group pairwise matrices and the compatibility between the individuals and group positions in four cases associated with two scenarios (weighted and non-weighted decision makers) and two situations (complete and incomplete PCCMs).

Two new methods for deriving the priority vector from interval multiplicative preference relations

Interval preference relations are widely used in the analytic hierarchy process (AHP) for their ability to express the expert’s uncertainty. The most crucial issue arises when deriving the interval priority vector from the interval preference relations. Based on two of the most commonly used prioritization methods (the eigenvalue method (EM) and the row geometric mean method (RGMM)), two new methods for obtaining the interval priority vector from interval multiplicative preference relations are developed, which endow the expert with different risk preferences for his/her interval judgments. In contrast to existing methods, new approaches calculate the interval priority weights of alternatives separately. Then, several concepts of acceptable consistency for interval multiplicative preference relations are defined. Using a convex combination method, the acceptable consistency of interval multiplicative preference relations can be derived from the associated and exact numerical relations. To increase the distinction of intervals, an improved interval ranking method is presented. After that, two algorithms that can cope with acceptably and unacceptably consistent cases are introduced. Meanwhile, three numerical examples are examined to show the application of the new approaches, and comparisons with several other methods are also made.

How can a group of procurement experts select suppliers? An approach for group decision support

Purpose – The application of theories on group decision support is yet to be explored extensively in supplier selection literature, although the literature in both domains is extremely rich, in isolation. The purpose of this paper is to explore the application of group decision support theories for supplier selection. Design/methodology/approach – The row geometric mean method (RGMM) of the analytic hierarchy process (AHP) has been used in this study for the prioritization of group preferences under consensus. A case study was conducted to test the theories of consensual group decision making and compare it with other approaches based on AHP. Findings – The study establishes that the application of decision support theories for group decision making can improve the supplier selection process. Findings further imply that RGMM is more effective than eigen value method, for group decision making under consensus. Research limitations/implications – Methodologically, the study highlights the greater regularity in outcome of group decision making, vis-à-vis individual decision making, for the same decision-making context. Also, it highlights how RGMM is more effective since it preserves reciprocal properties and diversity in preferences better. Practical implications – The study establishes that firms can improve supplier selection processes by leveraging on the collective expertise of a group rather than depending on individual decision-making expertise. Originality/value – This study explores the application of different theories based on AHP for consensual group decision making. It compares different approaches based on AHP and establishes that RGMM is a superior approach for supplier selection.

The precise consistency consensus matrix in a local AHP-group decision making context

A new decisional tool, the consensus consistency matrix (CCM) has recently been proposed for dealing with AHP-group decision making (AHP-GDM) in a local context (a single criterion). Each entry of this matrix, based on the property of consistency, corresponds to the range of values or interval in which all the decision makers are simultaneously consistent in their initial matrices. The main limitation of the CCM is that, on many occasions, it is not possible to obtain a matrix with the minimum $$n-1$$ judgments that are required to derive the priorities. In this local AHP context, using the row geometric mean as the prioritisation procedure, this paper presents an extension of the CCM, the precise consensus consistency matrix (PCCM), which significantly mitigates this problem. By identifying precise values in the common consistency intervals, the PCCM automatically allows the number of entries in the CCM to be increased. The PCCM provides more informed and participative GDM and offers more accurate estimations for the group’s priorities. It can also be used as a starting point for posterior negotiation processes between the actors and it can be employed in global AHP-GDM contexts (hierarchies). The new decisional tool has been applied to a real-life experience concerned with the analysis of the integral viability of public investment projects, more specifically, the economic valuation of social aspects.

An ILOWG operator based group decision making method and its application to evaluate the supplier criteria

The aim of this work is to present some cases of the induced linguistic ordered weighted geometry (ILOWG) operators and study their desired properties, which are very suitable to deal with group decision making (GDM) problems involving multiplicative linguistic preference relations. First, the concepts of compatibility index (CI) for two multiplicative linguistic preference relations are defined. Then, we provide some ILOWG operators to aggregate multiplicative linguistic preference relations in GDM problems. In particular, we present the compatibility index ILOWG (CI-ILOWG) operator, which induces the order of argument values by utilizing the compatibility index of experts; and the importance ILOWG (I-ILOWG) operator, which induces the order of argument values based on the importance index of the experts. Next, the reciprocity, consistency and compatibility properties of the collective multiplicative linguistic preference relations obtained by these cases of ILOWG operators are verified. Finally, the aggregation of individual judgements (AIJ) and the aggregation of individual priorities (AIP) provide the same priorities of alternatives by utilizing the row geometric mean method (RGMM) as a prioritization procedure and the ILOWG operators as an aggregation procedure. Our results show that if all the individual decision makers have an acceptable consensus degree, then the collective preference relation is also of an acceptable consensus degree. Moreover, the compatibility index induced linguistic ordered weighted geometric mean complex judgement matrix (CI-ILOWGCJM) guarantees that the group compatibility degree is at least as good as the arithmetic mean of all the individual compatibility degrees. Accordingly, a theoretic basis has been developed for the application of these cases of ILOWG operators in linguistic group decision making. Finally, a numerical example for evaluating criteria of supply selection is given to illustrate the application of the results.

Consensus models for AHP group decision making under row geometric mean prioritization method

The consistency measure is a vital basis for consensus models of group decision making using preference relations, and includes two subproblems: individual consistency measure and consensus measure. In the analytic hierarchy process (AHP), the decision makers express their preferences using judgement matrices (i.e., multiplicative preference relations). Also, the geometric consistency index is suggested to measure the individual consistency of judgement matrices, when using row geometric mean prioritization method (RGMM), one of the most extended AHP prioritization procedures. This paper further defines the consensus indexes to measure consensus degree among judgement matrices (or decision makers) for the AHP group decision making using RGMM. By using Chiclana et al.'s consensus framework, and by extending Xu and Wei's individual consistency improving method, we present two AHP consensus models under RGMM. Simulation experiments show that the proposed two consensus models can improve the consensus indexes of judgement matrices to help AHP decision makers reach consensus. Moreover, our proposal has two desired features: (1) in reaching consensus, the adjusted judgement matrix has a better individual consistency index (i.e., geometric consistency index) than the corresponding original judgement matrix; (2) this proposal satisfies the Pareto principle of social choice theory.