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The role of an expert in the decision-making process is crucial. If we ask an expert to help us to make a decision we assume their honesty. But what if the expert is dishonest? Then, the answer on how difficult it is for an expert to provide manipulated data in a given case of decision-making process becomes essential. In the presented work, we consider manipulation of a ranking obtained by the Geometric Mean Method applied to a pairwise comparisons matrix. More specifically, we propose an algorithm for finding an almost optimal way to swap the positions of two selected alternatives in a ranking. We also define a new index which measures how difficult such manipulation is in a given case. [ABSTRACT FROM AUTHOR]

Relative pairwise comparisons represent the cornerstone for several decision-making methods. Such approaches aim to support complex decision-making situations with multiple alternatives and are essential in order to provide an overall absolute evaluation of the alternatives despite the presence of experts and/or decision-makers with conflicting opinions. Moreover, when decision-maker's opinions are affected by uncertainty, there is the need to analyse the effect of such uncertain measures on the result of the decision-making process. We propose an approach based on a multi-objective optimisation problem, able to identify the presence of rank reversal issues in order to evaluate the stability of the final outcome of the decision-making process and a metric able to support experts in evaluating the effects of their uncertainty. We characterise the robustness of the ranking with respect to rank reversal by identifying a perturbation that is as small as possible, while causing the maximum number of ordinal swaps. [ABSTRACT FROM AUTHOR]

The aim of composite indicators is to express the overall performance of countries/regions with respect to a complex goal including sustainability, competitiveness, and innovation. Some of the indices play an important role in real governmental and strategic decisions on allocating sources. Sensitivity analyses usually include the changes in weights (of importance), the evaluations with respect to the criteria and the aggregating functions. In contrast, we investigate the effect of setting the minimal and maximal thresholds of the scoring functions used in the assessment. Thus, only the effect of this transformation is investigated, while the input data and criteria weights are not modified or stochastic. It is demonstrated that even such a seemingly innocent modification of the min/max thresholds might lead to remarkable changes in the ranking. Results are presented in detail on the examples of the Environmental Performance Index (EPI) and the Digital Economy and Society Index (DESI). However, the phenomenon is general: further 15 composite indices, applying the min/max threshold, have also been collected. The choice of min/max threshold is functionating as an implicit (re-)weighting of the criteria: criteria with smaller min/max ranges are overweighted. Thus, the steps of weighting and assessment are not independent. This research provides an alternative sensitivity analysis to test the robustness of the rankings.

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Abstract Financial performance analysis is of vital importance those involved in a business (e.g., shareholders, creditors, partners, and company managers). An accurate and appropriate performance measurement is critical for decision-makers to achieve efficient results. Integrated performance measurement, by its nature, consists of multiple criteria with different levels of importance. Multiple Criteria Decision Analysis (MCDA) methods have become increasingly popular for solving complex problems, especially over the last two decades. There are different evaluation methodologies in the literature for selecting the most appropriate one among over 200 MCDA methods. This study comprehensively analyzed 41 companies traded on the Borsa Istanbul Corporate Governance Index for 10 quarters using SWARA, CRITIC, and SD integrated with eight different MCDA method algorithms to determine the position of Turkey's most transparent companies in terms of financial performance. In this study, we propose "stock returns" as a benchmark in comparing and evaluating MCDA methods. Moreover, we calculate the "rank reversal performance of MCDA methods". Finally, we performed a "standard deviation" analysis to identify the objective and characteristic trends for each method. Interestingly, all these innovative comparison procedures suggest that PROMETHEE II (preference ranking organization method for enrichment of evaluations II) and FUCA (Faire Un Choix Adéquat) are the most suitable MCDA methods. In other words, these methods produce a higher correlation with share price; they have fewer rank reversal problems, the distribution of scores they produce is wider, and the amount of information is higher. Thus, it can be said that these advantages make them preferable. The results show that this innovative methodological procedure based on 'knowledge discovery' is verifiable, robust and efficient when choosing the MCDA method.

This work aims to answer the natural question of how probable it is that a given method produces rank reversal in a priority vector (PV) if a decision maker (DM) introduces perturbations to the pairwise comparison matrix (PCM) under concern. We focus primarily on the concept of robustness against rank reversal, independent of specific methods, and provide an in-depth statistical insight into the application of the Monte Carlo (MC) approach in this context. This concept is applied to three selected methods, with a special emphasis on scenarios where a method may not provide outputs for all possible PCMs. All results presented in this work are replicable using our open-source implementation. [ABSTRACT FROM AUTHOR]

So far, the rank reversal (RR) phenomenon has been heavily discussed in the multiple criteria analysis literature. In this work, instead of focusing on RR occurrences within PROMETHEE II, an analysis of the third alternative with respect to a given pair of alternatives is conducted. The aim is to provide a way to identify third alternatives able to cause RR between a specific pair. This opens the question of building alternatives causing RR occurrences. Such alternatives can then be used to assess the resistance a given ranking has to RR. In the end, two different approaches are used. The first considers the number of third alternatives to be added to generate RR for a given pair of alternatives. While, the second one, provides a new threshold for RR occurrences in PROMETHEE II. [ABSTRACT FROM AUTHOR]

Financial performance analysis is of vital importance those involved in a business (e.g., shareholders, creditors, partners, and company managers). An accurate and appropriate performance measurement is critical for decision-makers to achieve efficient results. Integrated performance measurement, by its nature, consists of multiple criteria with different levels of importance. Multiple Criteria Decision Analysis (MCDA) methods have become increasingly popular for solving complex problems, especially over the last two decades. There are different evaluation methodologies in the literature for selecting the most appropriate one among over 200 MCDA methods. This study comprehensively analyzed 41 companies traded on the Borsa Istanbul Corporate Governance Index for 10 quarters using SWARA, CRITIC, and SD integrated with eight different MCDA method algorithms to determine the position of Turkey's most transparent companies in terms of financial performance. In this study, we propose "stock returns" as a benchmark in comparing and evaluating MCDA methods. Moreover, we calculate the "rank reversal performance of MCDA methods". Finally, we performed a "standard deviation" analysis to identify the objective and characteristic trends for each method. Interestingly, all these innovative comparison procedures suggest that PROMETHEE II (preference ranking organization method for enrichment of evaluations II) and FUCA (Faire Un Choix Adéquat) are the most suitable MCDA methods. In other words, these methods produce a higher correlation with share price; they have fewer rank reversal problems, the distribution of scores they produce is wider, and the amount of information is higher. Thus, it can be said that these advantages make them preferable. The results show that this innovative methodological procedure based on 'knowledge discovery' is verifiable, robust and efficient when choosing the MCDA method. [ABSTRACT FROM AUTHOR]

Multiple criteria decision-making methods have been extremely useful for identifying the optimal alternative in complicated decision problems. However, in dynamic decision situations, they often suffer from rank reversal problem (RRP). For minimising the RRP, various improvements in existing methods have emerged in recent times. The proximity-indexed value (PIV) method is one method that promises to minimize the rank reversal problem. However, a detailed investigation of the RRP issue for the PIV method does not exist. In this paper, the PIV method has been taken up for a detailed examination of RRP using a MATLAB simulation-based framework by generating a sample of 10,000 random decision matrices. Seven approaches have been used to detect the existence of rank reversal. It is found that a sizeable prevalence of rank reversal remains in the PIV method. A new, improved PIV (IPIV) method is also proposed in this paper, which has been demonstrated to be effective in reducing the prevalence of rank reversal to almost zero. The effect of varying the number of criteria and the number of alternatives is also evaluated while considering the variation in criteria weights, i.e. equal and unequal criteria weights. Analysis of variance using the concept of blocking has been carried out on the results obtained at a 5% level of significance. The study also identifies more effective approaches from the seven approaches used to detect rank reversal. In addition, the application of the IPIV method is demonstrated in a real-world problem. [ABSTRACT FROM AUTHOR]

This paper aims to prove that when developing an inventory model with fuzzy demand, the order quantity should be identical for each replenishment cycle in order to achieve an optimal replenishment policy. In 2012, Glock, Schwindl and Jaber published a paper in the International Journal Services and Operations Management. They assumed that the order quantities should be identical for each replenishment cycle. We provide a patch work to prove that to attain the minimum solution, then the order quantities must be identical for each replenishment cycle. Moreover, a formulated solution for the optimal replenishment number is derived in this paper. On the other hand, we discuss two related problems in Analytic Hierarchy Process to present some theoretical derivations to help researchers realize the mathematical structure in those problems. [ABSTRACT FROM AUTHOR]

This paper presents a new multi‐criteria decision‐making (MCDM) method, namely the ratio product model (RPM). We first overview two popular aggregating models: the weighted sum model (WSM) and the weighted product model (WPM). Then, we argue that the two models suffer from some fundamental issues mainly due to ignoring the ratio nature of the alternatives' scores with respect to the criteria and the importance weights of the criteria. Building on the notion of compositional data analysis, the developed RPM regards performance scores and criteria weights as compositions, which solves the issues around the WSM and WPM. Using several examples, we show that the WSM and WPM could lead to erroneous conclusions, whereas the RPM could lead to fully reliable conclusions. Since many MCDM methods rely on some aggregation approaches, the proposed method is a significant contribution to the field and puts forward the correct way to analyze decision problems while respecting the nature and constraints of the input data. [ABSTRACT FROM AUTHOR]

Varied extensions of preference relations have been developed to cope with uncertain scenes. However, the introduced uncertainty, which might decrease the reliability of a final decision, has not been paid sufficient attention. In this study, a new concept, referred to as complex preference relation (CPR), is proposed to formalize most of the existing types of preference relations under uncertainty. Two distinct uncertainty measures are proposed to restrict the uncertainty degree of a CPR. The preference-oriented measure, which looks into the uncertainty of each piece of preference, gives limits to a CPR by its maximum variance. The rank-oriented measure, which focuses on the resultant ranks, restricts uncertainty by the probability of rank reversal. Stochastic uncertainty degrees are defined as a reference to enable the determination of uncertainty thresholds by risk appetites directly. A series of simulation experiments are conducted to demonstrate the features and connections of the two measures. [ABSTRACT FROM AUTHOR]

In recent times, appropriate decision-making in challenging and critical situations has been very well supported by multicriteria decision-making (MCDM) methods. The technique for order of preference by similarity to ideal solution (TOPSIS) is the most widely used MCDM method for solving decision problems. However, it restricts decision-makers to use only one type of Quality of Service (QoS) information, and it suffers from the rank reversal problem. Restriction to only one type of QoS makes the decision problems more challenging, as it restricts the decision-makers freedom. Further, the rank reversal problem makes the decision result unreliable. To address these issues of TOPSIS, we have proposed a reliable rank reversal robust modular TOPSIS (RMo-TOPSIS). RMo-TOPSIS allows crisp, interval, fuzzy, intuitionistic and neutrosophic fuzzy QoS metrics. It does not suffer from the rank reversal problem. Cloud computing provides computing services on-demand basis without involving maintenance by its users. The availability of many cloud service providers and their services makes cloud service selection a challenging problem. To validate RMo-TOPSIS, we select the cloud service selection consisting of different types of QoS metrics as an application. Experiments on cloud service selection show consistency and accuracy in results obtained by RMo-TOPSIS and its robustness against rank reversal. [ABSTRACT FROM AUTHOR]

When it comes to choosing the best option among multiple alternatives with criteria of different importance, it makes sense to use multi criteria decision making (MCDM) methods with more than 200 variations. However, because the algorithms of MCDM methods are different, they do not always produce the same best option or the same hierarchical ranking. At this point, it is important how and according to which MCDM methods will be compared, and the lack of an objective evaluation framework still continues. The mathematical robustness of the computational procedures, which are the inputs of MCDM methods, is of course important. But their output dimensions, such as their capacity to generate well-established real-life relationships and rank reversal (RR) performance, must also be taken into account. In this study, we propose for the first time two criteria that confirm each other. For this purpose, the financial performance (FP) of 140 listed manufacturing companies was calculated using nine different MCDM methods integrated with step-wise weight assessment ratio analysis (SWARA). İn the next stage, the statistical relationship between the MCDM-based FP final results and the simultaneous stock returns of the same companies in the stock market was compared. Finally, for the first time, the RR performance of MCDM methods was revealed with a statistical procedure proposed in this study. According to the findings obtained entirely through data analytics, Faire Un Choix Adéquat (FUCA) and (which is a fairly new method) the compromise ranking of alternatives from distance to ideal solution (CRADIS) were determined as the most appropriate methods by the joint agreement of both criteria.

Evaluating knowledge workers performance has always been one of the main concerns of knowledge-based organizations. The purpose of this article is to present two new versions of the additive ratio assessment (ARAS) method called Augmented ARAS (A-ARAS1 and A-ARAS2) that improve the rank reversal phenomenon of ARAS using some iterative methods to evaluate the optimal performance of knowledge workers. The study population consisted of the Academic Center for Education, Culture, and Research in Mashhad, Iran, which has been contributing to national development through the expansion of knowledge and applied research since 1995. The performance appraisal index was weighted using the best-worst method and then analyzed by the ARAS, A-ARAS1, and A-ARAS2. The results indicated that the A-ARAS1 and A-ARAS2 models outperformed the previous version of ARAS in terms of the performance appraisal of knowledge workers and that the ARAS model needed to be revised to improve performance appraisal. [ABSTRACT FROM AUTHOR]

In order to scientifically evaluate the emergency management capacity of major cities in China, this paper studies the input and output index data of cities in emergency management. Firstly, the urban emergency management evaluation system is established according to the emergency investment funds, the proportion of emergency investment funds in GDP, the number of emergency personnel, the number of accidents, the number of deaths, and the loss of funds and so on. Then, a super-efficiency DEA urban emergency management capability evaluation model based on the weight constraint of the order relation analysis method (G1 method) is constructed to achieve a reasonable combination of subjective and objective assignments. Finally, the emergency management data of 36 major cities in China are selected for effectiveness evaluation, and the cities with the most effective emergency input relative to accident output are obtained, and the rank reversal problem existing in the traditional DEA model is discussed. The results show that the super efficiency DEA with weight constraint of G1 method proposed in this paper not only considers the subjective information of decision-makers, but also realizes the objective assignment, which can effectively solve the problem of rank reversal and achieve the effective and credible ranking of urban emergency management ability. [ABSTRACT FROM AUTHOR]

Sustainable material selection is a crucial problem given the new demands of society and novel production strategies that consider the concepts of sustainability. Multi-criteria decision-making methods have been extensively used to help decision-makers select alternatives in different fields of knowledge. Nonetheless, these methods have been criticized due to the rank reversal problem, where the independence of the irrelevant alternative principle is violated after the initial decision problem is changed. Over the course of this study, we observed that the solutions that are proposed for this problem, in the context of sustainable material selection, are insufficient. Thus, we present a new material selection approach that is based on the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method, which is immune to rank reversal. We also demonstrate the causes of rank reversal in the TOPSIS method, how the R-TOPSIS method was designed to solve them, and how it can be applied to sustainable material selection. [ABSTRACT FROM AUTHOR]

In the current account, we present an analysis of a non-discriminating criterion under simple additive weighting synthesis, considering a deep decision hierarchy. A non-discriminating criterion describes a criterion where all decision alternatives under consideration perform equally. We question eliminating such a criterion from the decision hierarchy in search of simpler problem representation and computational efficiency. Yet, we prove such an approach may result in order misrepresentations between decision alternatives. This analysis is performed in the form of four research questions that relate to the detection of certain conditions under which such distortions in the order integrity of decision alternatives will occur, calculating the change in their final performances, distinguishing the alternatives whose performances are consistent, and examining the role of the normalization procedure adopted in averting such distortions when the non-discriminating criterion is ignored. Along these lines, this study provides clear inferences which are of interest to researchers and decision makers, using simple additive weighting and similar methods that rely on additive synthesis. [ABSTRACT FROM AUTHOR]

Promethee ii is a multi‐criteria decision aid method working as follows: firstly, all alternatives are compared two by two to form a pairwise comparison matrix. Then the net flow score procedure is applied on this matrix to assign a score to each alternative according to which they are ranked. Methods of the Promethee family may suffer from rank reversal occurrences. The legitimacy of methods suffering from this phenomenon has been largely debated in the literature. The aim of this work is to provide a characterization of the Promethee ii net flow scores as a combination between two distinct methods based on different hypotheses. The first one is based on the direct pairwise comparison of the two considered alternatives while the second is based on the comparison with respect to all the other alternatives of the problem. This combination highlights the dependence on third alternatives as an inherent characteristic of the Promethee ii method. [ABSTRACT FROM AUTHOR]

The goal of this work is to apply Multiple Criteria Decision Analysis tools, both theoretical and practical, to analyse, support and possibly enhance composite indexes, in particular those related to sustainability assessment. In this context, the Sustainable Development of Energy, Water and Environment Systems Index represents a paradigmatic example and an emerging reference point, thus it is specifically addressed throughout the work. On the theoretical side, the focus is on the property of “independence”, i.e., of evaluating an alternative independently of the others. It is argued that this property can be appealing for an index that is conceived to address, over time, an increasing number of inherently evolving systems. A viable and theoretically grounded approach for devising a version of the index fulfilling independence is proposed. On the practical side, the contribution concerns visual support tools. A well-known projective method is adapted to work with the index, and a new tool with comparable expressive capabilities is proposed. The new representation is more focused on the index, technically simpler, and less sensitive to changes in the input data. The features of the visual tools are illustrated exploiting currently available (partially aggregated) index data. In particular, the new tool is used to illustrate two issues addressed in the scientific literature on the index, namely, the use of scenario analysis as a predictive tool, and the decoupling of energy usage and carbon dioxide emissions.

This work proposes a Fuzzy Proximity Index Ranking method to facilitate multi-criteria decision making in fuzzy environment. The proposed method is derived from proximity index value approach by incorporating the fuzzy concept to take care of vagueness in the information provided by the experts. The method is implemented on several multi-criteria decision-making problems taken from the literature to investigate its performance. Ranking results of the proposed method are found closer to those of Fuzzy TOPSIS, Fuzzy MOORA, Fuzzy VIKOR and Fuzzy ELECTRE. Further, a few case studies on rank reversal are presented on three different fuzzy scenarios to examine the robustness of the method. The results revealed that the proposed method suffers least from the rank reversal issue and it generates reasonably accurate ranking order compared to other techniques. Hence, the proposed method is considered a simple fuzzy based multi-criteria decision approach which can be a reliable substitute for existing methods. [ABSTRACT FROM AUTHOR]

Multiple attribute decision-making (MADM) methods are commonly employed to assist decisions for selecting the best alternative according to conflicting criteria in complex decision situations. Although MADM methods have proven to be very useful, in some dynamic decision cases they may cause faulty results due to the "rank reversals" problem. Numerous MADM methods such as Analytic Hierarchy Process (AHP), Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), Preference Ranking Organization Method for Enrichment Evaluations (PROMETHEE), and several others well-known methods proven to have rank reversal problems. In this paper, we study the rank reversal problem for a recent MADM method that is known as Weighted Aggregated Sum Product Assessment (WASPAS). As far as we know, there is no study that considers the rank reversal problem in the WASPAS. In this paper, rank reversal problems in WASPAS are analyzed empirically by considering different types of rank reversals. After detailed computational experiments, we show that rank reversal problems also exist in WASPAS when classical normalization techniques are utilized. We also show through extensive computational experiments by using different problem instances that rank reversal problems can be avoided in WASPAS when modified Max and Max–Min normalization techniques are used. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to improve the classical VIKOR method with a holistic view, and evaluate the alternatives with invariant reference points and scales to achieve the rank preservation. The VIKOR method, as a commonly used multi-attribute decision making model, is prone to rank reversal when the alternative is added, deleted or replaced. The phenomenon of rank reversal indicates that there is a contradiction in the consistency and credibility of the evaluation results. First, the new improved VIKOR called R-VIKOR focuses on the selection of historical extreme values of indicators, the process of data normalization, and the determination of virtual ideal solutions. It should be emphasized that the improved method is the improvement and extension of VIKOR. In addition, the theoretical basis of R-VIKOR with the ability to preserve rank is discussed. The R-VIKOR can ensure that the evaluation result of the selected alternative can be fixed, which is just like a unified ruler measuring objects, and the problem of rank reversal can be solved naturally. Finally, several examples and random experiments are examined to verify the validity and credibility of R-VIKOR.

The Analytic Hierarchy Process (AHP) is a decision making method, which has as its greatest criticism the rank reversal effect. Here a new mathematical analysis of this method is performed, and three new results are highlighted. First, the method is formulated as a linear system of equations, where it is possible to assign a geometric interpretation, determine the number of possible solutions, and perform an sensitivity analysis based on the condition number of the matrix. Second, the causes of rank reversal can be encompassed by two mathematical aspects related to the properties of the matrix of AHP: high condition number and deficient rank. When the matrix is deficient rank, it is possible to obtain a condensed formulation of the AHP with a new full rank matrix. This guarantees greater stability to the method. Third, some mathematical results can be used as a robustness test for the matrix of AHP. [ABSTRACT FROM AUTHOR]

VIKOR is one of the well known Multiple Criteria Decision Aid methods, used frequently and considered to produce robust results. However, it seems that it is not immune to the rank reversal phenomenon. The latter is a well studied phenomenon that occurs sometimes in many such methods and is the cause of often severe criticism. This short paper provides such an example (and a counter-example) in the case of a fuzzy version of VIKOR.

In the current account, we present an analysis of a non-discriminating criterion under simple additive weighting synthesis, considering a deep decision hierarchy. A non-discriminating criterion describes a criterion where all decision alternatives under consideration perform equally. We question eliminating such a criterion from the decision hierarchy in search of simpler problem representation and computational efficiency. Yet, we prove such an approach may result in order misrepresentations between decision alternatives. This analysis is performed in the form of four research questions that relate to the detection of certain conditions under which such distortions in the order integrity of decision alternatives will occur, calculating the change in their final performances, distinguishing the alternatives whose performances are consistent, and examining the role of the normalization procedure adopted in averting such distortions when the non-discriminating criterion is ignored. Along these lines, this study provides clear inferences which are of interest to researchers and decision makers, using simple additive weighting and similar methods that rely on additive synthesis.

Several multiple criteria decision making(A/ICI)M) techniques have been used to assist decision-makers (DMs) in selecting better alternatives for various problems. However, it has been observed that with the addition of new alternatives or the deletion of existing ones, the rank of available alternatives will present a problem referred to as rank reversal. An improved grey relational analysis method is proposed in which the information expansion method and virtual ideal scheme are used to prevent changes of extreme value in standardization when there are alterations in the alternative set. These are the main reasons for rank reversal. When comparing literature on results from case studies and simulated cases, it is clear that the new method can maintain a robust rank of alternatives. This indicates that the proposed method is capable of preventing the rank reversal phenomenon, which arises out of changes in available alternatives. [ABSTRACT FROM AUTHOR]

Logistics management has been playing a significant role in ensuring competitive growth of industries and nations. This study proposes a new Multi-Criteria Decision-making (MCDM) framework for evaluating operational efficiency of logistics service provider (LSP). We present a case study of comparative analysis of six leading LSPs in India using our proposed framework. We consider three operational metrics such as annual overhead expense (OE), annual fuel consumption (FC) and cost of delay (CoD, two qualitative indicators such as innovativeness (IN) which basically indicates process innovation and average customer rating (CR)and one outcome variable such as turnover (TO) as the criteria for comparative analysis. The result shows that the final ranking is a combined effect of all criteria. However, it is evident that IN largely influences the ranking. We carry out a comparative analysis of the results obtained from our proposed method with that derived by using existing established frameworks. We find that our method provides consistent results; it is more stable and does not suffer from rank reversal problem. [ABSTRACT FROM AUTHOR]

In the current study, based on grey relational similarity measure and entropy-weight, a comprehensive grey relational similarity measitie method for multi-attribute decision making (MADM) problem to evaluate the alternatives has been proposed. A new algorithm based on the same attribute value's comparison and different attribute value's synthesis for the MADM problems is also established. In this algorithm, the ideal attribute values are selected according to attribute properties. Later, the grey relational similarity measure method is used to calculate the grey value between the attribute value and the ideal attribute value in the same attribute. To compare the alternatives, the entropy-weight method is applied to determine the weights for different attributes objectively. Finally, we take the multi-attribute weapon system problem as an example to illustrate the effectiveness and validity of the proposed method. The example verification restilts show that the method constructed in this paper has higher credibility, and the rank reversal prnhlem lirts nlfn passed the test. [ABSTRACT FROM AUTHOR]

Rank reversal in Analytic Hierarchy Process (AHP) due to introduction or removal of a new alternative is well documented in literature. However, rank reversal due to the aggregation of benefit and cost criteria has not been addressed with requisite rigour. This paper demonstrates a new type of rank reversal in AHP which can arise due to the method (ratio or difference) and approach (sum 1 or sum 2) used to aggregate the benefit and cost criteria. Numerical examples, mathematical analyses and computer simulations have been used for demonstration of results. It is found that considering the benefit and cost criteria separately (sum 2 approach) while aggregating them can yield irrational ranking. It is also demonstrated that ratio method of aggregation is untenable in additive variants of AHP as it yields identical ranking in sum 1 and sum 2 approaches. Difference method of aggregation considering the benefit and cost criteria together (sum 1 approach) is most logical and sound in additive variants of AHP. The results also counter the notion that multiplicative AHP is immune to rank reversal between ratio and difference methods of aggregation. Besides, sum 1 and sum 2 approaches produce different raking of alternatives in multiplicative AHP irrespective of the method of aggregation. Outcome of this research will be helpful while choosing the appropriate method and approach for aggregation of benefit and cost criteria in different variants of AHP.

Multicriteria decision making (MCDM) methods are widely used in decision making practice. However, these methods are heuristic: they are based on some rules, the effectiveness and validity of which cannot be strictly proved, and it is unknown exactly how trustworthy their outputs reflect Decision-maker's judgements. Such methods can generate different outputs, using the same inputs, and it is impossible to select the more correct method from the available ones – this is the subject of the decision-making paradox. In this article, we focus on the possibility to construct a non-heuristic MCDM method, whose accuracy and reliability can be verified. We base the proposed method on two new ideas on how to construct a pairwise comparison (PC) matrix for a multi-criteria decision-making problem with several criteria and alternatives, and how to correctly derive information from this matrix. We show that it is possible (i) to estimate the objective uncertainty in the PC-matrix information with respect to each alternative and (ii) to estimate the reliability of the rankings. The non-heuristic MCDM method based on these ideas allows to verify the accuracy and reliability of Decision-maker's judgements on MCDM problems, and has clear anti-reversal guarantees. The software implementing the method is freely available online at . [ABSTRACT FROM AUTHOR]

The analytic hierarchy process is a widely used multi-criteria decision-making method that involves the construction of pairwise comparison matrices. To infer a decision, a consistent or near-consistent matrix is desired, and therefore, several methods have been developed to control or improve the overall consistency of the matrix. However, controlling the overall consistency does not necessarily prevent having strong local inconsistencies. Local inconsistencies are local distortions which can lead to rank reversal when a new alternative is added or deleted. To address this problem, we propose an algorithm for controlling the inconsistency during the construction of the pairwise comparison matrix. The proposed algorithm assists decision makers whilst entering their judgments and does not allow strong local inconsistencies. This algorithm is based on the transitivity rule and has been verified through statistical simulations. Appropriate thresholds of acceptable evaluations have been inferred from these simulations. We demonstrate that the proposed algorithm is a helpful decision aid to decision makers when entering pairwise comparison judgments. [ABSTRACT FROM AUTHOR]