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An Examination of Ranking Quality for Simulated Pairwise Judgments in relation to Performance of the Selected Consistency Measure

An Examination of Ranking Quality for Simulated Pairwise Judgments in relation to Performance of... Hindawi Advances in Operations Research Volume 2019, Article ID 3574263, 24 pages https://doi.org/10.1155/2019/3574263 Research Article An Examination of Ranking Quality for Simulated Pairwise Judgments in relation to Performance of the Selected Consistency Measure Paul Thaddeus Kazibudzki Universite Internationale Jean-Paul II de Bafang, B.P. 213, Bafang, Cameroon Correspondence should be addressed to Paul aTh ddeus Kazibudzki; emailpoczta@gmail.com Received 16 October 2018; Revised 7 December 2018; Accepted 24 December 2018; Published 3 February 2019 Academic Editor: Eduardo Fernandez Copyright © 2019 Paul Thaddeus Kazibudzki. is Th is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An overview of current debates and contemporary research devoted to modeling decision making processes and their facilitation directs attention to techniques based on pairwise judgments. At the core of these techniques are various judgment consistency measures which, in a sense, control the prioritization process which leads to the establishment of decision makers’ unknown preferences. If judgments expressed by decision makers were perfectly consistent (cardinally transitive), all available prioritization techniques would deliver the same solution. However, human judgments are consistently inconsistent, as it were; thus the preference estimation quality significantly varies. The scale of these variat ions depends, among others, on the chosen consistency measure of pairwise judgments. That is why it seems important to examine relations among various consistency measures and the preferences estimation quality. This research reveals that there are consis tency measures whose performance may confuse decision makers with the quality of their ranking outcome. u Th s, it introduces a measure which is directly related to the quality of the preferences estimation process. The main problem of the research is studied via Monte Carlo simulations executed in Wolfram Mathematica Sow ft are. The research results argue that although the performance of examined consistency measures deviates from the exemplary ones in relation to the estimation quality of decision makers preferences, solutions proposed in this paper can significantly improve that quality. phenomena and relating them by making comparisons. The 1. Introduction latter method leads directly to the essence of the matter, Overwhelming scientific evidence indicates that the unaided i.e., judgments regarding a phenomenon. Judgments can be human brain is simply not capable of simultaneous analysis of relative or absolute. An absolute judgment is the relation many dieff rent, competing factors and then synthesizing the between a single stimulus and some information held in short results for the purpose of making a rational decision. Indeed, or long term memory. A relative judgment, on the other hand numerous psychological experiments, e.g., [1], including the [3], is defined as identification of some relation between two well-known Miller [2] study put forth the notion that humans stimuli both present to the observer. Certainly humans can are not capable of dealing accurately with more than about make much better relative than absolute judgments. u Th s, a pairwise comparisons method was proposed in order to seven (±2) things at a time (the human brain is limited in its short term memory capacity, its discrimination ability, and its facilitate the process of relative judgments. range of perception). Some authors proclaim [4, 5] that this method dates back to the beginning of the 20th century and was firstly applied Humans learn about anything by two means: the rfi st involves examining and studying some phenomenon from by u Th rstone [6]; however, its rs fi t scienticfi applications can the perspective of its various properties and then synthe- be found in Fechner [7]. In reality, the method itself is much sizing findings and drawing conclusions; the second entails older and its idea goes back to Ramon Lull who lived in the studying some phenomenon in relation to other similar end of 13th century. It is a fact that its popularity comes from 2 Advances in Operations Research an influential paper of Marquis de Condorcet [8]; see, e.g., as “Compare – applying a given ratio scale – your feelings [9, 10], who used this method in the election process where concerning alternative 1 versus alternative 2”, do not produce voters rank candidates based on their preference. It has been accurate outcomes. us, Th A(𝑤 )isnot established but only its perfected in many papers, e.g., [4, 11–20]. estimate A(x) containing intuitive judgments, more or less Also, the method is the core of a decision making close to A(𝑤 ) in accordance with experience, skills, specific methodology—the Analytic Hierarchy Process (AHP), devel- knowledge, personal taste, and even temporary mood or oped by Saaty [21]. Although it is criticized (see [22–25, overall disposition. In such case, consistency property does 25–28]) it is also continuously perfected and as such very not hold and the relation between elements of A(x)and A(𝑤 ) oeft n validated [29] and developed from the perspective can be expressed as𝑥 =𝑒 𝑤 where𝑒 is a perturbation factor of its utility (see [5, 30–37]). u Th s, considering the stable u fl ctuating near unity. In the statistical approach, e reefl cts increase of the AHP application (see, e.g., [38–44]) as well a realization of a random variable with a given probability as its steady position among other theories of choice, e.g., distribution. [31–37, 45–47], the problematic issues of AHP should be Besides the prioritization procedure (PP) proposed by the continuously examined and questions which arise thereof creator of AHP, right principal eigenvector method (REV), should be simultaneously addressed. there are alternative PPs devised to cope with the priority The fundamental objective of this research is to determine ratios estimation problem; their demonstrative review can be the answer to the question: Does the reduction of PCM incon- found, e.g., in [47]. Many of them are optimization based sistency lead to improvement of the priority ratios estimation and seek a vector 𝑤 , as a solution of the minimization process quality? problem given by the formula {min𝐷 (𝐴 (𝑥), 𝐴 (𝑤))} subject to some assigned constraints, such as positive coefficients and normalization condition. Because the distance function 2. Background D measures an interval between matrices A(x)and A(𝑤 ), The AHP uses the hierarchical structure of the decision different den fi itions of the distance function lead to vari- problem, pairwise relative comparisons of the elements in ous prioritization concepts and prioritization results. As an the hierarchy, and a series of redundant judgments which example, eighteen PPs in [48] are described and compared enable measurement of judgment consistency. To make a for ranking purposes although some authors suggest there proposed solution possible, i.e., derive ratio scale priorities on are only fteen fi that are different. Furthermore, since the the basis of verbal judgments, a scale is utilized to evaluate publication of the above-mentioned article, a few additional the preferences for each pair of items. Probably, the most procedures have been introduced to the literature; see, e.g., known scales are Saaty’s numerical scale which comprises [43, 49–51]. Probably the most popular alternative to the REV integers, and their reciprocals, from one (equivalent to the is the Logarithmic Least Squares Method (LLSM) developed verbal judgment, “equally preferred”) to nine (equivalent to by Crawford and Williams [26, 52]. It is given by the following the verbal judgment, “extremely preferred”), and a geometric formula: scale which usually consists of the numbers computed in 𝑛 𝑛 𝑛/2 𝑤 = min∑ ∑ ln (𝑎 ) accordance with the formula𝑓(𝑛) = 𝑐 for 𝑛 ∈ {−8,...,8}∧ (1) ()𝑀 𝑖=1 𝑗=1 𝑛∈ 𝐼 where c denotes its parameter which commonly equals 2. Other arbitrarily den fi ed numerical scales are also The LLSM solution also has the following closed form and is available, e.g., composed of arbitrary integers from one to n given by the normalized products of the elements in each row: and their reciprocals. 1/𝑛 The key issue in AHP is priority ranking on the basis of (∏ 𝑎 ) 𝑗=1 𝑤 = (2) true or approximate weights, i.e., judgments. If the relative 𝑖()𝑀𝑆𝐿𝐿 1/𝑛 𝑛 𝑛 ∑ (∏ 𝑎 ) weights of a set of activities are known, they can be expressed 𝑖=1 𝑗=1 as a Pairwise Comparison Matrix (PCM): A(𝑤 )=(𝑤 /𝑤 ),i, 𝑖 𝑗 u Th s, it is also known as the geometric mean method j=1,..., n. PCM in the AHP reflects decision makers’ prefer- anditisutilized inthisresearch which strives to improve the ences (their relative judgments) about considered activities reliability of the pairwise comparisons process which is also (criteria, scenarios, players, alternatives, etc.). On the basis of the core element of AHP. A(𝑤 ),itispossibletoderivetrueweights;i.e.,decision makers priority ratios 𝑤 ,where: i=1,...,n, are selected to be positive 3. Problem and Research Methodology and normalized to unity: ∑ 𝑤 =1. For uniformity, 𝑤 is 𝑖 𝑖 referred hereaer ft to its normalized form. If the elements There are several PCM consistency measures (PCM-CMs) of a matrix A(𝑤 ) satisfy the condition 𝑤 =1/𝑤 for all i, 𝑗𝑖 provided in literature called consistency or inconsistency j=1,..., n then matrix A(𝑤 )is called reciprocal. If the elements indices (CIs). The most popular one is the PCM-CM pro- of a matrix A(𝑤 ) satisfy the condition 𝑤 𝑤 =𝑤 for all i, posed by Saaty [21]. He proposed his CI as determined by the j, k=1,..., n,and thematrix is reciprocal, then it is called formula consistent or cardinally transitive. 𝜆 −𝑛 max Certainly, in real life situations when AHP is utilized, (3) 𝑛−1 there is no A(𝑤 ) which would reflect weights given by the vector of priority ratios. As was stated earlier, the human where n indicates the number of alternatives within the mind is not a reliable measurement device. Assignments, such particular PCM and𝜆 denotes its maximal eigenvalue. The max 𝑅𝐸𝑉 𝐶𝐼 𝑖𝑗 𝑘𝑗 𝑖𝑘 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝐿𝐿𝑆 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 Advances in Operations Research 3 significant disadvantage of the PCM-CM is the fact that it – Two formulae for its measurement: 󵄨 󵄨 can operate exclusively with reciprocal PCMs. In the case of 𝑎 𝑎 󵄨 󵄨 𝑗𝑘 󵄨 󵄨 󵄨 󵄨 𝐼𝐿𝑇 (𝑎 ,𝑎 ,𝑎 )=󵄨 ln 󵄨 (8) nonreciprocal PCMs, this measure is useless (its values are 1 𝑗𝑘 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 meaningless) which in consequence seriously diminishes the 𝑎 𝑎 value of the whole approach; see, e.g., [33]. It was also recently 𝑗𝑘 𝐼𝐿𝑇 (𝑎 ,𝑎 ,𝑎 )= ln (9) 2 𝑗𝑘 found to be incorrect; see, e.g., [4, 10, 53, 54]. However, as mentioned earlier, there are a number of – One meaningful formula for PCM-CM: additional PCM-CMs. Some of them, as in the case of CI , REV originate from the PPs devised for the purpose of the priority 𝑛−2 𝑛−1 𝑛 ratios estimation process. Their distinct feature is the fact that 𝐼𝑇 (𝐴 )= ∑ ∑ ∑ {𝐿𝐼𝑇 (𝑎 ,𝑎 ,𝑎 )} (10) 𝑥 𝑛 𝑥 𝑗𝑘 ( ) all of them can operate equally efficiently in conditions where 𝑖=1 𝑗=𝑖+1 𝑘=𝑗+1 reciprocal and nonreciprocal PCMs are accepted. Probably where x denotes the formula for triad inconsistency the most known example from that set of propositions is measurement, i.e., LTI or LTI . 1 2 PCM-CM proposed by Aguaron and Moreno-Jimenez [55] It behooves us to mention that ALTI(A)can be calculated given by the following formula: on the basis of triads from the upper triangle of the given 𝑎 𝑤 PCM when it is reciprocal or all triads within the given PCM = ∑ log ( ) 𝑀 (4) when it is nonreciprocal. (𝑛−1 )(𝑛− 2 ) 𝑤 𝑖<𝑗 As was already stated earlier, the fundamental question which should be asked by researchers who deal with the Noticeably, there are a few den fi itions of PCM-CMs problem of priority ratios estimation quality in relation to a which are not connected with any PP and are devised on the PCM consistency measure is as follows: Does the reduction of basis of the PCM consistency definition. Koczkodaj’s idea [56] PCM inconsistency lead to improvement of the priority ratios attracts attention and is the rfi st to be scrutinized. Koczkodaj’s estimation process quality? PCM-CM is grounded in his concept of triad consistency. The common reason why one strives to improve the In order to clarify this, for any three distinguished deci- consistency of the PCM, when it seems unsatisfactory, is to sion alternatives A ,A ,and A , there are three meaningful 1 2 3 increase the quality of the priority ratios estimation process. priority ratios, i.e., a , a ,and a , which have their different 𝑗𝑘 However, the above question remains open and the answer locations in a particular 𝐴 =[𝑎 ] .For some dieff rent i≤n, 𝑛×𝑛 to it is not evident. Even the creator of AHP stated once j≤n, and k≤n, the tuple (a , a , a )is called a triad. If the 𝑗𝑘 that improving consistency does not mean getting an answer matrix 𝐴 =[𝑎 ] is consistent, then a a = a for all 𝑛×𝑛 𝑗𝑘 closer to the “real” life solution [21]. It can be illustrated in the triads. following example. In consequence, either of the equations 1− 𝑎 /𝑎 𝑎 = 𝑗𝑘 Considered is the true PV (denoting true weights of 0 and 1−𝑎 𝑎 /𝑎 =0 have to be true. Taking the above 𝑗𝑘 examined alternatives), i.e., 𝑤 =[7/20, 1/4, 1/4, 3/20] and A(𝑤 ) into consideration, Koczkodaj proposed his measure for triad derived from that PV, which can be presented as follows: inconsistency by the following formula: 7 7 7 󵄨 󵄨 1 󵄨 󵄨 󵄨 󵄨 𝑎 𝑎 [ ] 󵄨 󵄨 󵄨 󵄨 𝑎 𝑗𝑘 5 5 3 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 [ ] (𝑎 ,𝑎 ,𝑎 )= min { 1− , 1− } (5) 󵄨 󵄨 󵄨 󵄨 𝑗𝑘 󵄨 󵄨 󵄨 󵄨 [ ] 󵄨 󵄨 󵄨 󵄨 𝑎 𝑎 𝑎 5 5 󵄨 󵄨 󵄨 󵄨 [ ] 𝑗𝑘 󵄨 󵄨 [ ] 7 3 [ ] 𝐴 (𝑤 )= (11) [ ] Following his idea, he then proposed the following PCM-CM [ ] 5 5 [ ] of any reciprocal PCM=A: [ ] 7 3 [ ] [ ] 3 3 3 𝐾 (𝐴 )= max{𝑇𝐼 (𝑎 ,𝑎 ,𝑎 )} 𝑛 1 𝑝 𝑗𝑘 (6) 𝑝=1,..., ( ) 𝑖<𝑗<𝑘 3 [ ] 7 5 5 Then two PCMs are considered, i.e., R(x)and A(x)pro- where the maximum value for K(A) is taken from the set of duced by a hypothetical decision maker (DM). It is assumed all possible triads in the upper triangle of a given PCM. that DM is very trustworthy and is able to express judgments On the basis of Koczkodaj’s idea of triad inconsistency, very precisely at the same time being still somehow limited by Grzybowski [5] presented his PCM-CM determined by the the necessity of expressing judgments on a scale (the example following formula: utilizes Saaty’s scale). In the rfi st scenario, entries of A(𝑤 )are rounded to Saaty’s scale and the entries are made reciprocal 𝐼 (𝐴 ) (a principal condition for a PCM in the AHP) producing 󵄨 󵄨 𝑛−2 𝑛−1 𝑛 󵄨 󵄨 󵄨 󵄨 (7) 𝑎 𝑎 󵄨 󵄨 󵄨 󵄨 1 𝑎 󵄨 󵄨 󵄨 𝑗𝑘 󵄨 󵄨 󵄨 󵄨 󵄨 11 1 2 = ∑ ∑ ∑ min { 󵄨 1− 󵄨 ,󵄨 1− 󵄨 } 𝑛 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 [ ] ( ) 𝑎 𝑎 𝑎 󵄨 󵄨 󵄨 󵄨 3 𝑗𝑘 𝑖=1 𝑗=𝑖+1 𝑘=𝑗+1 󵄨 󵄨 [ ] 11 1 2 [ ] [ ] 𝑅 (𝑥 )= (12) 11 1 2] Finally, following the idea that ln(𝑎 /𝑎 𝑎 )=−ln(𝑎 𝑎 / 𝑗𝑘 𝑗𝑘 [ ] [ ] 𝑎 ), Kazibudzki [57] redefined triad inconsistency and pro- 1 1 1 posed [ 2 2 2 ] 𝑖𝑘 𝑖𝑗 𝑖𝑗 𝑖𝑘 𝑖𝑘 𝑖𝑗 𝑖𝑘 𝑖𝑗 𝐴𝑇 𝑖𝑘 𝑖𝑗 𝑖𝑘 𝑖𝑗 𝑖𝑘 𝑖𝑗 𝑇𝐼 𝑖𝑘 𝑖𝑗 𝑖𝑘 𝑖𝑗 𝑖𝑗 𝑖𝑘 𝑖𝑘 𝑖𝑗 𝑖𝑗 𝑖𝑘 𝑖𝑗 𝑖𝑗 𝑖𝑘 𝑖𝑗 𝐿𝐿𝑆 𝐶𝐼 𝑖𝑗 𝑖𝑘 𝑖𝑗 𝐴𝐿 𝑖𝑘 𝑖𝑘 𝑖𝑗 𝑖𝑗 𝑖𝑘 𝑖𝑘 𝑖𝑗 𝑖𝑗 4 Advances in Operations Research Table 1: Values of the PCM-CMs denoted as CI(PP )for R(x) and proposed characteristics of PVs estimates (∗) quality in relation to the true PV for the case. Performance measures PP Estimates CI(PP)MAE SRC REV [0.285714,0.285714,0.285714,0.142857] 0.0 0.0357143 0.8164966 LLSM [0.285714,0.285714,0.285714,0.142857] 0.0 0.0357143 0.8164966 (∗): derived from R(x) with application of a particular PP. Table 2: Values of the PCM-CMs denoted as CI(PP )for A(x) and proposed characteristics of PVs estimates (∗) quality in relation to the true PV for the case. Performance measures PP Estimates CI(PP)MAE SRC REV [0.309401,0.267949,0.267949,0.154701] – 0.0893164 0.0202995 1 LLSM [0.314288,0.264284,0.264284,0.157144] 0.0400378 0.0178559 1 (∗): derived from A(x) with application of a particular PP. In the second scenario, only entries of A(𝑤 ) are rounded to is very true in the situation when the particular PCM is Saaty’s scale (nonreciprocal case) producing less consistent (on the basis of selected exemplary PCM- CMs). It remains to be mentioned that the inconsistency 11 1 2 problem (hence errors) does not exist for simplified pairwise [ ] comparisons [53]. [ ] [ ] 11 2 Taking into consideration only such a trivial example like [ ] [ 2 ] the one above, it becomes apparent that the relation between [ ] 𝐴 (𝑥 )= (13) [ ] performance of a consistency measure and the quality of [ ] 11 2 [ ] the priority ratios estimation process is of great importance. [ ] [ ] 1 1 1 That is why to examine the phenomenon further to improve the quality of the pairwise comparisons based prioritization [ ] 2 2 2 process was decided. u Th s, the simulation framework for It should be noted thatR(x)is perfectly consistent and this purpose was adopted from [5, 57] as the only way to A(x) is not. Tables 1 and 2 present selected values of the examine the said phenomena through computer simulations. PPs related PCM-CMs (that is, CI and CI )forR(x) REV LLSM The simulation algorithm SA|K| is comprised of the following and A(x)together with PVs derived from R(x)and A(x); phases. Mean Absolute Errors (MAEs), formula (14), among 𝑤∗ (PP) and 𝑤 for the case; Spearman Rank Correlation Coecffi ients Phase 1. Randomly generate a priority vector 𝑤 =[𝑤 , (SRCs) among 𝑤∗ (PP) and 𝑤 for the case. ...,𝑤 ] of assigned size [n x1] and related perfect PCM(𝑤 )=K(𝑤 ). 󵄨 󵄨 󵄨 󵄨 𝐸 (𝑤∗ ( ),𝑤 )= ∑ 𝑤 −𝑤 ∗ ( ) 󵄨 󵄨 (14) 󵄨 𝑖 𝑖 󵄨 Phase 2. Randomly select an element𝑤 for x<y ofK(𝑤 )and 𝑖=1 replace it with 𝑤 𝑒 where e is a relatively significant error, 𝐵 𝐵 Surprisingly, a very interesting phenomenon can be noted randomly drawn (uniform distribution) from the interval on the basis of information provided in Tables 1 and 2. The 𝑒 ∈ [2;4]. Errors of that magnitude are basically considered nonreciprocal version of the analyzed PCM contains nonzero “significant”; see, e.g., [50, 58]. values for the selected PCM-CMs. In cases similar to this example, the value of Saaty’s PCM-CM always becomes neg- Phase 3. For each other element 𝑤 , i<j≤n, select randomly a ative which makes it inexplicable and in consequence useless value e for the relatively small error in accordance with the under such circumstances (as already mentioned earlier). The given probability distribution 𝜋 (applied in equal proportions other exemplary measure taken into consideration is positive as gamma, log-normal, truncated normal,and uniform distri- and higher values than zero which indicates that particular bution) and replace the element 𝑤 with the element 𝑤 𝑒 PCM is inconsistent. On the basis of the same indicators in where e is randomly drawn (uniform distribution) from the the case of the reciprocal version of the analyzed PCM, its interval 𝑒 ∈ [0,5;1,5]. perfect consistency is apparent because all selected PCM- CMs in this case are equal to zero. However, the estimation Phase 4. Round all values of 𝑤 𝑒 for i<j ofK(𝑤 )to the precision measures (MAE and SRC), i.e., characteristics of nearest value of a considered scale. the particular PV estimation quality, indicate something quite opposite. Surprisingly, smaller values of MAEs are apparent, Phase 5. Replace all elements 𝑤 for i>j ofK(𝑤 )with 1/𝑤 . as are perfect correlations of ranks between estimated and Phase 6. After all replacements are done, return the value of genuine PV for nonreciprocal version of the analyzed PCM. Certainly, this conclusion concerns all analyzed PPs and it the examined index as well as the estimate of the vector 𝑤 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑥𝑦 𝑥𝑦 𝑃𝑃 𝑃𝑃 𝑀𝐴 Advances in Operations Research 5 denoted as 𝑤 ∗(PP) with application of assigned prioritiza- which would contradict such a pertinent relationship would tion procedure. Then return the mean absolute error (MAE) unequivocally lead to the conclusion that the examined PCM- and mean relative error (MRE) (formula (15)) between 𝑤 CM is not a good indicator of the priority ratios estimation and 𝑤 ∗(PP). Remember values computed in this phase as process quality and may mislead in further actions towards one record. acceptance or rejection of the derived priority ratios vector. 󵄨 󵄨 𝑛 An examination of the research problem depicts Figures 󵄨 󵄨 𝑤 −𝑤 ∗ 1 󵄨 ( )󵄨 𝑖 𝑖 󵄨 󵄨 𝐸 (𝑤∗ ( ),𝑤 )= ∑ (15) 1 and 2 which present the performance of Koczkodaj K(A) 𝑛 𝑤 𝑖=1 (Plots A–H). In each case hereafter a horizontal axis rep- resents values of examined PCM-CM, and a vertical axis Phase 7. Repeat Phases from Phase 2 to Phase 6 N times. denotes particular estimation errors. Noticeably, when the quality of priority ratios estimation Phase 8. Repeat Phases from Phase 1 to Phase 7 N times. in a pairwise comparisons based process is taken into consid- Phase 9. Save all records as one database file. eration, the presented relations indicate that the performance of selected PCM-CM varies from the assumption presented The above algorithm allows examining the performance earlier. The phenomenon was thus far only examined in [5]. of the selected CM in relation to the quality of the priority It may seem disturbing because the relation between priority ratios estimation process. Its framework resembles steps ratios estimation errors and analyzed PCM-CM indicates that scrutinized in the example provided earlier in this paper and the analyzed index may sometimes misinform DMs about was thoroughly described in [5]. u Th s, for brevity, it will not their judgments’ quality, affecting the vector of priorities be analyzed in detail herein. which best converge with the true vector. For formality, all parameters of the applied PDs, gamma, As seen similarly in the example provided earlier in this log-normal, truncated normal,and uniform,in the simulation paper (Tables 1 and 2), taking the particular index as the algorithm SA|K| are set in such a way that the expected measure of PCM consistency, especially when errors are small value EV(e )=1. The simulation begins from n=4. Simulations and inconsistency is not negligible, one can expect both, for n=3 are not interesting due to direct interrelation of i.e., the improvement of priority ratios estimation quality considered PCM consistency measures; see, e.g., [58, 59]. For (increase of the estimation accuracy) together with the the sake of objectivity, the simulation data is gathered in the increase of the particular CI (decrease of PCM consistency) following way: all values of selected consistency measures and, inversely, the deterioration of priority ratios estimation are split into 15 separate sets designated by the quantiles Q quality (decrease of the estimation accuracy) together with of order p from 1/15 to 14/15. The 15 intervals are defined the descent of the particular CI (improvement of PCM as the rfi st is from 0 to the quantile of order 1/15, i.e., consistency). VRCM =[0, 𝑄 ), where VRCM represents aValue Range This problem seems very troublesome especially when 1 1/15 of the Selected PCM Consistency Measure; the second denotes differences among derived priority ratios are insignificant. VRCM =[𝑄 , 𝑄 ), and so on... to the last one which Then, it may occur that a ranking order of alternatives in 2 1/15 2/15 starts from the quantile of order 14/15 and goes on to infinity, the estimated priority vector can drastically differ from the i.e., VRCM =[𝑄 ,∞). The following variables are true one because of estimation errors. From that perspective, 15 14/15 examined: mean VRCM ,average MAEwithin VRCM a necessity of controlling that issue seems paramount. In n n between 𝑤 and 𝑤∗ (PP), MAE quantiles of the following order to evaluate that problem more thoroughly, detailed orders, 0.05, 0.1, 0.5, 0.9, and 0.95, and relations between all statistical characteristics for the examined K(A)are provided of them. The application of the rounding procedure was also in Tables 3–6. It can be concluded that the performance of the assumed which in this research operates according to Saaty’s presented PCM-CM varies (compare, e.g., [9]). scale. Lastly, the scenario takes into account the compulsory The motivation for this research was to develop a PCM- assumption in conventional AHP applications, i.e., the PCM CM which could depict a more credible relation between reciprocity condition. The outcome of the simulation pro- the consistency of pairwise judgments and the priority ratios gram is presented for the most popular PP=LLSM, and the estimation quality, i.e., whose priority ratios estimation error, most attractive PCM-CM=K(A). It must be emphasized that reflected by SRC, would be very close or equal to 1 for all their other PPs and PCM-CMs were also examined but results with quantiles (the most desirable situation). their application will not be presented in this research paper Successfully, a solution of the problem was generated and because thisisbeyond itsscope. The resultsare based on is presented as follows. On the basis of triad inconsistency N =20, and N =500, i.e., 10,000 cases. measureintroduced in[57], thefollowing PCM-CM was 𝑛 𝑚 devised: 4. Results and Discussion (𝐴 ) 𝑛−2 𝑛−1 𝑛 2 One could assume that MAE and MRE quantiles of any ∑ ∑ ∑ ln (𝑎 /𝑎 𝑎 ) 𝑗𝑘 (16) 𝑖=1 𝑗=𝑖+1 𝑘=𝑗+1 order should monotonically increase concurrently with the = growth of the selected PCM-CM, e.g., VRCM index. The ( )(1 + max {ln (𝑎 /𝑎 𝑎 )} ) 3 𝑖<𝑗<𝑘 𝑝 𝑗𝑘 𝑝=1,..., ( ) same relation should occur for mean VRCM and average MAE and MRE for VRCM . The results of the proposed sim- The proposed PCM-CM is denoted as the Triads Squared ulation framework or any other similar simulation scenario Logarithm Corrected Mean and an examination of its 𝑖𝑗 𝑖𝑘 𝑖𝑗 𝑖𝑘 𝑇𝑆𝐿 𝑖𝑗 𝑃𝑃 𝑀𝑅 𝑃𝑃 6 Advances in Operations Research 0.07 Plot |A| for n=4 Plot |B| for n=4 0.020 MSRC= 0.889286 MSRC= 0.975 0.06 Quantile(0.05) Median 0.05 0.015 0.04 0.010 0.03 0.02 0.005 0.01 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.08 Plot |C| for n=4 Plot |D| for n=4 0.15 MSRC= 0.971429 MSRC= 0.985714 Quantile(0.95) Mean 0.06 0.10 0.04 0.05 0.02 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.05 0.020 Plot |E| for n=6 Plot |F| for n=6 MSRC= 0.942857 MSRC= 0.942857 0.04 Quantile(0.05) Median 0.015 0.03 0.010 0.02 0.005 0.01 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.10 0.05 Plot |G| for n=6 Plot |H| for n=6 MSRC= 0.935714 MSRC= 0.967857 0.08 0.04 Quantile(0.95) Mean 0.06 0.03 0.02 0.04 0.01 0.02 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 1: Performance of K(A)as PCM-CM. eTh plots present the relation between mean values of K(A) withinagiveninterval (VRCM ) and values of MAEs and quantiles of a given order for MAEs distribution concerning estimated and true PV for the case. The results are 𝑛/2 generated with application of LLSM as the PP and geometric scale, i.e., 𝑓(𝑛) = 𝑐 for 𝑛 ∈ {−8,...,8} ∧ 𝑛 ∈ 𝐼 as the preference scale with its most popular parameter c=2. Plots are based on 10,000 random reciprocal PCMs for n=4 and n=6. MSRC denotes MeanSpearmanRank Correlation Coefficient between analyzed variables and describes their relation’s strength. Advances in Operations Research 7 0.35 0.12 Plot |A| for n=4 Plot |B| for n=4 0.30 MSRC= 0.832143 MSRC= 0.960714 0.10 Quantile(0.05) 0.25 Median 0.08 0.20 0.06 0.15 0.04 0.10 0.02 0.05 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.6 Plot |C| for n=4 Plot |D| for n=4 1.5 MSRC= 0.939286 MSRC= 0.953571 0.5 Quantile(0.95) Mean 0.4 1.0 0.3 0.2 0.5 0.1 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.15 0.35 Plot |E| for n=6 Plot |F| for n=6 0.30 MSRC= 0.903571 MSRC= 0.953571 Quantile(0.05) Median 0.25 0.10 0.20 0.15 0.05 0.10 0.05 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.7 1.5 0.6 Plot |G| for n=6 Plot |H| for n=6 MSRC= 0.935714 MSRC= 0.771429 0.5 Quantile(0.95) Mean 1.0 0.4 0.3 0.5 0.2 0.1 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 2: Performance of K(A)as PCM-CM. eTh plots present the relation between mean values of K(A)within a given interval(VRCM ) and values of MREs and quantiles of a given order for MREs distribution concerning estimated and true PV for the case. The results are 𝑛/2 generated with application of LLSM as the PP and geometric scale, i.e., 𝑓(𝑛) = 𝑐 for 𝑛 ∈ {−8,...,8} ∧ 𝑛 ∈ 𝐼 as the preference scale with its most popular parameter c=2. Plots are based on 10,000 random reciprocal PCMs for n=4 and n=6. MSRC denotes MeanSpearmanRank Correlation Coefficient between analyzed variables and describes their relation’s strength. 8 Advances in Operations Research 𝑖 𝑝 Table 3: Performance of the K(A)index. Statistical characteristics of the MAEs distribution in relation to various VRCM for i=1,...,15 of K(A) values. The results were generated for n=4 on the basis of SA|K| as the simulation algorithm and are based on 10,000 perturbed random reciprocal PCMs. The scenario assumed LLSM as the PP and Saaty’s preference scale. –quantiles of MAEs among and 𝑤∗ (LLSM) VRCM for 𝐾() Mean 𝐾() in VRCM Average MAEs among and𝑤∗ (LLSM) = 0.05 = 0.1 = 0.5 = 0.9 = 0.95 1 [0.00,0.293) 0.123175 0.003832 0.005244 0.013085 0.031933 0.044036 0.017332 2 [0.293,0.343) 0.298548 0.004178 0.005586 0.014531 0.039371 0.057014 0.020352 3 [0.343,0.394) 0.367621 0.007113 0.009216 0.020061 0.055438 0.077366 0.027509 4 [0.394,0.444) 0.418743 0.008836 0.011492 0.024425 0.068702 0.089414 0.033247 5 [0.444,0.494) 0.468803 0.010929 0.014093 0.031328 0.080087 0.097035 0.040523 6 [0.494,0.544) 0.505368 0.006643 0.009028 0.025446 0.076131 0.099659 0.035366 7 [0.544,0.595) 0.570415 0.014658 0.019158 0.044794 0.086016 0.104479 0.049809 8 [0.595,0.645) 0.621502 0.018761 0.023914 0.047224 0.082449 0.099615 0.051410 9 [0.645,0.695) 0.660045 0.012166 0.016648 0.042915 0.089487 0.109722 0.048732 10 [0.695,0.746) 0.721332 0.020838 0.025263 0.046327 0.085889 0.101378 0.051714 11 [0.746,0.796) 0.765848 0.017353 0.022895 0.049089 0.093422 0.110678 0.054968 12 [0.796,0.846) 0.822092 0.020521 0.026287 0.054130 0.093059 0.109786 0.058010 13 [0.846,0.897) 0.871055 0.020469 0.026452 0.053883 0.095374 0.114356 0.058507 14 [0.897,0.947) 0.921011 0.019656 0.026032 0.056166 0.111384 0.132945 0.063610 15 [0.947, oo) 0.969329 0.020548 0.027742 0.070211 0.142947 0.168917 0.078853 Advances in Operations Research 9 𝑖 𝑝 Table 4: Performance of the K(A)index. Statistical characteristics of the MREs distribution in relation to various VRCM for i=1,...,15 of K(A) values. The results were generated for n=4 on the basis of SA|K| as the simulation algorithm and are based on 10,000 perturbed random reciprocal PCMs. The scenario assumed LLSM as the PP and Saaty’s preference scale. –quantiles of MREs among and𝑤∗ (LLSM) VRCM for 𝐾() Mean 𝐾() in VRCM Average MREs among and𝑤∗ (LLSM) = 0.05 = 0.1 = 0.5 = 0.9 = 0.95 1 [0.00,0.293) 0.122019 0.021006 0.028171 0.061569 0.146142 0.218195 0.096958 2 [0.293,0.343) 0.298608 0.023159 0.029979 0.071009 0.192293 0.314784 0.117319 3 [0.343,0.394) 0.367711 0.041275 0.049211 0.098279 0.296609 0.400347 0.155107 4 [0.394,0.444) 0.418546 0.049447 0.060524 0.119874 0.366732 0.524257 0.271243 5 [0.444,0.495) 0.469288 0.064006 0.075775 0.162260 0.402196 0.521439 0.282957 6 [0.495,0.545) 0.505474 0.036757 0.048691 0.127198 0.403209 0.575937 0.238749 7 [0.545,0.596) 0.571548 0.086091 0.109329 0.227951 0.454215 0.609114 0.354275 8 [0.596,0.646) 0.622419 0.123841 0.152670 0.226833 0.451058 0.627101 0.348887 9 [0.646,0.697) 0.660911 0.070502 0.096137 0.214043 0.511722 0.707955 0.349459 10 [0.697,0.747) 0.722936 0.135092 0.147302 0.217057 0.425921 0.548016 0.303217 11 [0.747,0.798) 0.767036 0.112457 0.134261 0.256527 0.514438 0.715117 0.377223 12 [0.798,0.848) 0.823550 0.122843 0.149805 0.262036 0.490951 0.662803 0.363241 13 [0.848,0.899) 0.872625 0.125868 0.158338 0.257350 0.518942 0.714696 0.377362 14 [0.899,0.949) 0.922412 0.123110 0.152628 0.284100 0.655613 0.988531 0.471624 15 [0.949, oo) 0.970250 0.123223 0.157524 0.362451 1.033090 1.822590 0.672675 10 Advances in Operations Research 𝑖 𝑝 Th Table 5: Performance of the K(A)index. Statistical characteristics of the MAEs distribution in relation to various VRCM for i=1,...,15 of K(A) values. The results were generated for n=6 on the basis of SA|K| as the simulation algorithm and are based on 10,000 perturbed random reciprocal PCMs. e scenario assumed LLSM as the PP and the geometric preference s cale. –quantiles of MREs among and𝑤∗ (LLSM) VRCM for 𝐾() Mean 𝐾() in VRCM Average MREs among and𝑤∗ (LLSM) = 0.05 = 0.1 = 0.5 = 0.9 = 0.95 1 [0.00,0.500) 0.309970 0.003138 0.003923 0.007291 0.013229 0.015591 0.008080 2 [0.500,0.537) 0.501271 0.004210 0.005171 0.010393 0.019240 0.023050 0.011567 3 [0.537,0.573) 0.554137 0.006699 0.008198 0.016071 0.029390 0.035490 0.017867 4 [0.573,0.610) 0.592713 0.007459 0.009298 0.018158 0.030929 0.035619 0.019454 5 [0.610,0.646) 0.630157 0.009699 0.011333 0.019057 0.030303 0.034814 0.020245 6 [0.646,0.683) 0.650693 0.006237 0.007816 0.015671 0.028959 0.034565 0.017455 7 [0.683,0.719) 0.702542 0.009187 0.010726 0.018481 0.030391 0.035464 0.019892 8 [0.719,0.756) 0.745895 0.008338 0.010188 0.019239 0.036328 0.043535 0.021702 9 [0.756,0.792) 0.774678 0.009249 0.010954 0.019613 0.034921 0.041489 0.021766 10 [0.792,0.829) 0.816861 0.010058 0.012214 0.023462 0.043198 0.051295 0.026023 11 [0.829,0.865) 0.847194 0.010501 0.012745 0.022585 0.038486 0.044885 0.024501 12 [0.865,0.902) 0.880103 0.011555 0.013988 0.026175 0.049286 0.058456 0.029391 13 [0.902,0.938) 0.921940 0.013008 0.016109 0.032457 0.059978 0.069519 0.035730 14 [0.938,0.975) 0.958146 0.014460 0.018227 0.037434 0.067413 0.078266 0.040733 15 [0.975, oo) 0.985451 0.019837 0.024678 0.049174 0.085952 0.097939 0.052645 Advances in Operations Research 11 Th Table 6: Performance of the K(A)index. Statistical characteristics of the MREs distribution in relation to various VRCM for i=1,...,15 of K(A) values. The results were generated for n=6 on the basis of SA|K| as the simulation algorithm and are based on 10,000 perturbed random reciprocal PCMs. e scenario assumed LLSM as the PP and the geometric preference s cale. p–quantilesof MAEsamong w and w∗(LLSM) VRCM for 𝐾() Mean 𝐾() in VRCM Average MAEs among w and w ∗ (LLSM) p=0.05 p=0.1 p=0.5 p=0.9 p=0.95 1 [0.00,0.500) 0.310089 0.023352 0.028073 0.048709 0.085736 0.100416 0.054715 2 [0.500,0.537) 0.501135 0.031293 0.037867 0.070485 0.130644 0.163143 0.146845 3 [0.537,0.573) 0.554356 0.053930 0.063208 0.116827 0.235816 0.318104 0.172693 4 [0.573,0.610) 0.592078 0.069405 0.084870 0.136276 0.226008 0.293496 0.249541 5 [0.610,0.646) 0.630342 0.087466 0.096573 0.132412 0.216063 0.282887 0.401898 6 [0.646,0.683) 0.650498 0.048560 0.059216 0.113700 0.235798 0.330528 0.226842 7 [0.683,0.719) 0.701886 0.078667 0.086119 0.125897 0.231228 0.292786 0.173261 8 [0.719,0.756) 0.746012 0.067538 0.079147 0.139081 0.326062 0.481375 0.227734 9 [0.756,0.792) 0.774741 0.073180 0.082412 0.139194 0.274548 0.366040 0.196296 10 [0.792,0.829) 0.817070 0.079380 0.094561 0.169309 0.395980 0.580617 0.273966 11 [0.829,0.865) 0.847198 0.088182 0.101460 0.153473 0.300013 0.418148 0.224248 12 [0.865,0.902) 0.880101 0.093595 0.108170 0.189377 0.449450 0.667343 0.305017 13 [0.902,0.938) 0.921891 0.105242 0.125400 0.246040 0.605379 1.034870 0.459726 14 [0.938,0.975) 0.958183 0.118728 0.145412 0.276527 0.581686 0.861373 0.433898 15 [0.975, oo) 0.985442 0.157070 0.190925 0.372889 0.888862 1.638710 0.686757 12 Advances in Operations Research 0.030 0.06 0.025 0.05 0.020 0.04 0.015 0.03 0.010 0.02 Plot |A| Plot |B| 0.005 0.01 MSRC=0.996429 MSRC=1 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.12 0.07 0.06 0.10 0.05 0.08 0.04 0.06 0.03 0.04 0.02 Plot |C| Plot |D| MSRC=1 MSRC=1 0.02 0.01 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 Figure 3: Performance of the new PCM-CM: TSL(A). eTh plots present the relation between a mean value of TSL(A)within a giveninterval (VRCM ) and quantilesof order 0.1 (Plot A), 0.5 (PlotB), and0.9 (PlotC) of MAEsdistribution aswell as MAEsaverage values (PlotD) for estimated and true PVs. The results were generated with application of LLSM as the PP and Saaty’s scale. Plots are based on 10,000 random reciprocal PCMs for n=4. performance on the basis of simulation algorithm SA|K| It is noted that all statistical characteristics of the MAEs proposed earlier in this paper was carried out (Figures 3 and and MREs distribution in relation to various VRCM for 4). i=1,...,15 of TSL(A) values, with few exceptions, monoton- As can be noticed, the proposed TSL(A)performs ically grow. This examination ascertains that the proposed credibly from the perspective of the relation among the PCM-CM is a suitable measure of relation among pairwise consistency of pairwise judgments and the priority ratios comparisons consistency and the priority ratios estimation estimation quality. It is undeniably a positive piece of infor- quality. The paramount position of the proposed TSL(A)is mation which opens a new chapter in pairwise judgments that it performs better than the other, evaluated here, PCM- based priority ratios estimation process embedded in many CM, i.e., K(A). Its position is additionally strengthened by the methodologies of decision making such as AHP. It behooves fact that its performance is similar and independent from the us to mention that TSL(A) is suitable for both reciprocal and applied PP and improves significantly for higher numbers of nonreciprocal PCMs which prospectively may improve the alternatives without regard to which PP is selected. pairwise judgments based priority ratios estimation quality It should be noted that all characteristics presented herein when nonreciprocal PCMs are accepted. are of great importance in the priority ratios estimation Tables 7 and 8 provide detailed characteristics data for process, because one has to consider the potential of rejecting TSL(A) with application of LLSM (as the most popular a “good” PCM, and vice versa, i.e., the possibility of accepting alternative to REV), Saaty’s scale, and geometric scale as the a “bad” PCM, as in the classic statistical hypothesis testing most popular preference scales. Results for other PPs are theory. However, for the rfi st time in the course of pairwise similar; thus they are not presented in order to conserve judgments based prioritization development history, the pos- the length of this paper. However, it is stressed that other sibility of selecting the level of certainty and basing decisions PPs and preference scales were also tested and examined on statistical facts has been demonstrated. during the research and their results are not depicted in For instance, considering some hypothetical PCM for this paper because they coincide with results herein pre- n=4, with its mean TSL(A)≈0.319702 for LLSM as the PP sented. (Table 7), one can expect with 95% confidence that the MAE Advances in Operations Research 13 0.15 0.30 0.25 0.10 0.20 0.15 0.05 0.10 Plot |B| Plot |A| 0.05 MSRC=0.996429 MSRC=0.978571 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.8 0.6 0.5 0.6 0.4 0.4 0.3 0.2 Plot |D| 0.2 Plot |C| 0.1 MSRC=0.996429 MSRC=1 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 Figure 4: Performance of the new PCM-CM: TSL(A). eTh plots present the relation between a mean value of TSL(A)within a giveninterval (VRCM ) and quantiles of order 0.1 (Plot A), 0.5 (Plot B), and 0.9 (Plot C) of MREs distribution as well as MREs average values (Plot D) for estimated and true PVs. eTh results were generated with application of LLSM as the PP and geometric scale. Plots are based on 10,000 random reciprocal PCMs for n=4. should not exceed the value of 0.1208420. At the same time, as previously. Assuming this time TSL(A)≈0.049049, it can be one can expect with 95% confidence that it will be higher anticipated with 95% certainty that the MAE<0.072271 which than 0.0201740 (Table 7). Whether one decides to accept rather insures that the order of alternatives ranks should such a PCM or reject it obviously depends on the quality remain unchanged. requirements of the priority ratios estimation and the attitude For similar but even more detailed calculation, MRE regarding these errors. Indeed, the outcome of the research can be applied. It is a more accurate measure of priority finally creates the potential for true consistency control in an ratios deviation; however, its straightforward application unprecedented way, i.e., directly related to the priority ratios for calculation of discrepancies within normalized priority estimation quality. vectors is problematic. For example, the following PV is considered as 𝑤 =[0.27, In order to enable other researchers to make similar 0.26,0.24,0.23] denoting DM preferences for alternatives A , analysis concerning different numbers of alternatives, the A ,A ,and A , respectively. Taking into consideration the 2 3 4 exemplary characteristics of TSL(A) performance are pro- earlier assumed level of TSL(A)≈0.319702, the order of alter- vided for n>4 in the Appendix of this article, computed with natives ranks A =1, A =2, A =3, and A =4 can be very decep- 1 2 3 4 application of commonly known PP, i.e., LLSM and the most tive and is rather meaningless. In such a situation one can common Saaty’s scale (Table 9) and geometric scale (Table 10) expect with 95% confidence that the MAE >0.0201740 which as prospective preferences scales selected by decision makers. makes one aware that the true rank order of examined prefer- ences may appear otherwise, due to estimation errors related to DM inconsistency, e.g., 𝑤∗ =[(0.27–0.025), (0.26–0.025), 5. Conclusions (0.24+0.025), (0.23+0.025)] = [0.242, 0.231, 0.269, 0.258], with MAE=0.025, which designates a different order for In this research, the performance of the selected PCM- alternatives ranks, A =3, A =4, A =1, and A =2. On the CM from the perspective of its relation between pairwise 1 2 3 4 other hand, consider PV as 𝑤 =[0.49,0.33,0.16,0.02] of DM judgments consistency and the quality of the priority ratios preferences for alternatives A ,A ,A ,and A consecutively, estimation process was examined with application of the most 1 2 3 4 14 Advances in Operations Research Table 7: Performance of the TSL(A)index. Statistical characteristics of the MAEs distribution in relation to various VRCM for i=1,...,15 of TSL(A) values. The results were generated for n=4 on the basis of SA|K| as the simulation algorithm and are based on 10,000 perturbed random reciprocal PCMs. The scenario assumed LLSM as the PP and Saaty’s preference scale. –quantiles of MAEs among and𝑤∗ (LLSM) VRCM for (𝑇𝑆 𝐴) Mean (𝑇𝑆 𝐴) in VRCM Average MAEs among and𝑤∗ (LLSM) = 0.05 = 0.1 = 0.5 = 0.9 = 0.95 1 [0.00,0.0934) 0.049049 0.005253 0.007007 0.017709 0.047828 0.072271 0.024607 2 [0.0934,0.12) 0.108368 0.006543 0.008965 0.022188 0.064636 0.089647 0.030783 3 [0.120,0.147) 0.131977 0.007560 0.010195 0.025401 0.073557 0.096264 0.034682 4 [0.147,0.173) 0.161289 0.009281 0.012408 0.032397 0.084857 0.106224 0.041842 5 [0.173,0.200) 0.186567 0.010605 0.014283 0.039235 0.087242 0.107040 0.046369 6 [0.200,0.227) 0.213651 0.012731 0.017479 0.044342 0.092117 0.112116 0.050310 7 [0.227,0.253) 0.240868 0.017165 0.022310 0.047478 0.093918 0.112928 0.053405 8 [0.253,0.280) 0.267645 0.018953 0.024106 0.048903 0.095909 0.112627 0.055422 9 [0.280,0.307) 0.293803 0.020081 0.025244 0.052348 0.097503 0.114723 0.058089 10 [0.307,0.333) 0.319702 0.020174 0.025936 0.054471 0.101461 0.120842 0.060164 11 [0.333,0.360) 0.345876 0.021179 0.027149 0.055049 0.104366 0.126714 0.061558 12 [0.360,0.387) 0.372744 0.021740 0.027979 0.056325 0.107628 0.130267 0.063053 13 [0.387,0.413) 0.400500 0.022274 0.028479 0.057966 0.110502 0.132659 0.064974 14 [0.413,0.440) 0.425325 0.021991 0.028264 0.060355 0.116391 0.137831 0.067429 15 [0.440, oo) 0.509413 0.022479 0.029361 0.063909 0.122018 0.144845 0.071126 𝐿𝐿𝑆 Advances in Operations Research 15 Th Table 8: Performance of the TSL(A)index. Statistical characteristics of the MREs distribution in relation to various VRCM for i=1,...,15 of TSL(A) values. The results were generated for n=4 on the basis of SA|K| as the simulation algorithm and are based on 10,000 perturbed random reciprocal PCMs. e scenario assumed LLSM as the PP and geometric scale for prefer ences. –quantiles of MREs among and𝑤∗ () VRCM for (𝑇𝑆 𝐴) Mean (𝑇𝑆 𝐴) in VRCM Average MREs among and𝑤∗ (LLSM) = 0.05 = 0.1 = 0.5 = 0.9 = 0.95 1 [0.00,0.063) 0.038493 0.021222 0.027957 0.063897 0.162196 0.250369 0.101603 2 [0.063,0.092) 0.075288 0.043098 0.049606 0.093880 0.275593 0.404725 0.159070 3 [0.092,0.120) 0.108927 0.030409 0.041118 0.100685 0.327138 0.441926 0.170932 4 [0.120,0.149) 0.131428 0.038427 0.049762 0.121698 0.379335 0.529179 0.199342 5 [0.149,0.177) 0.165031 0.054076 0.070318 0.180633 0.449114 0.626229 0.285351 6 [0.177,0.206) 0.194473 0.051717 0.069238 0.199856 0.462612 0.640247 0.301912 7 [0.206,0.234) 0.220825 0.089354 0.120756 0.223531 0.469640 0.630734 0.320947 8 [0.234,0.263) 0.250293 0.108335 0.138489 0.218002 0.465311 0.629102 0.324638 9 [0.263,0.291) 0.277319 0.114172 0.135641 0.246244 0.504231 0.682016 0.356697 10 [0.291,0.320) 0.306276 0.120962 0.143374 0.258090 0.507218 0.698669 0.361393 11 [0.320,0.348) 0.333741 0.122310 0.147515 0.260439 0.552231 0.775952 0.390815 12 [0.348,0.377) 0.361859 0.123757 0.156182 0.263475 0.579019 0.842248 0.425032 13 [0.377,0.405) 0.390045 0.133595 0.160033 0.267660 0.619568 0.920619 0.443646 14 [0.405,0.434) 0.416203 0.126431 0.153950 0.289740 0.699219 1.141290 0.523514 15 [0.434, oo) 0.506589 0.129256 0.162990 0.328123 0.834585 1.430670 0.632685 16 Advances in Operations Research Th 𝑖 𝑖 Table 9: Performance of TSL(A) index under the action of LLSM as the PP. Statistical characteristics of the MAEs distribution in relation to various levels of TSL(A) withina givenVRCM for i=1,...,15. The results are based on 10,000 perturbed random reciprocal PCMs with application of Saaty’s scales and were generated on the basis of SA|K| as the simulation algorithm. e table contains results for ∈ {5,6,7,8,9} , presented consecutively. –quantiles of MAEs among w and w∗(LLSM) VRCM for (𝑇𝑆 𝐴) Mean (𝑇𝑆 𝐴) in VRCM Average MAEs among w and w ∗ (LLSM) =0.05 =0.1 =0.5 =0.9 =0.95 1 [0, 0.0899) 0.057912 0.0039186 0.0049954 0.0109799 0.0221753 0.0274887 0.0127898 2 [0.0899,0.107) 0.099124 0.0056158 0.0073876 0.0157136 0.0324243 0.0398139 0.0183201 3 [0.107,0.124) 0.116088 0.0063140 0.0079525 0.0184299 0.0389673 0.0490687 0.0214159 4 [0.124,0.142) 0.133907 0.0075132 0.0102233 0.0230429 0.0443459 0.0539668 0.0258898 5 [0.142,0.159) 0.151127 0.0099921 0.0129535 0.0261046 0.0486044 0.0581258 0.0290851 6 [0.159,0.176) 0.167911 0.0113191 0.0142543 0.0289546 0.0558904 0.0682777 0.0328936 7 [0.176,0.193) 0.184671 0.0125612 0.0158052 0.0320054 0.0594491 0.0730399 0.0357402 8 [0.193,0.211) 0.201896 0.0136853 0.0171375 0.0339101 0.0640703 0.0789391 0.0380755 9 [0.211,0.228) 0.219329 0.0142803 0.0178080 0.0361548 0.0711273 0.0839402 0.0408705 10 [0.228,0.245) 0.236371 0.0150518 0.0185369 0.0380656 0.0762136 0.0919801 0.0435024 11 [0.245,0.262) 0.253302 0.0161087 0.0208189 0.0405464 0.0789105 0.0929572 0.0462684 12 [0.262,0.280) 0.270523 0.0160427 0.0205586 0.0431223 0.0821647 0.0965329 0.0482168 13 [0.280,0.297) 0.288211 0.0165698 0.0209757 0.0457022 0.0865715 0.100490 0.0504072 14 [0.297,0.314) 0.305099 0.0177870 0.0226112 0.0455671 0.0859316 0.100544 0.0507868 15 [0.314, oo) 0.357080 0.0186614 0.0241816 0.0493007 0.0932224 0.107664 0.0547348 p–quantiles of MAEs among w and w∗(LLSM) i VRCM for TSL(A)Mean TSL(A)inVRCM Average MAEs among w and w ∗ (LLSM) p=0.05 p=0.1 p=0.5 p=0.9 p=0.95 1 [0, 0.0901) 0.0618775 0.0036042 0.0044511 0.0090509 0.0175099 0.0212066 0.0102634 2 [0.0901,0.102) 0.096694 0.00584585 0.0072472 0.0147877 0.0255157 0.0297767 0.0158427 3 [0.102,0.115) 0.109186 0.0071783 0.0088408 0.0167774 0.0304119 0.0360603 0.0186253 4 [0.115,0.127) 0.121228 0.00831565 0.0102091 0.0192100 0.0349865 0.0420209 0.0214601 5 [0.127,0.139) 0.133028 0.0088771 0.0109435 0.0208206 0.0393504 0.0481357 0.0236802 6 [0.139,0.151) 0.144977 0.0097898 0.0118208 0.0225534 0.0439163 0.0538868 0.0259512 7 [0.151,0.163) 0.156874 0.0101678 0.0126009 0.0248914 0.0500528 0.0613696 0.0288113 8 [0.163,0.176) 0.169306 0.0113233 0.0138144 0.0274455 0.0552421 0.0656847 0.0317599 9 [0.176,0.188) 0.181783 0.0120341 0.0147276 0.0297646 0.0587297 0.0700824 0.0339487 10 [0.188,0.200) 0.193745 0.0124796 0.0157621 0.0317564 0.0613300 0.0720410 0.0356610 11 [0.200,0.212) 0.205758 0.0137977 0.0167981 0.0329443 0.0622977 0.0721443 0.0368687 12 [0.212,0.225) 0.218204 0.0140878 0.0175574 0.0347152 0.0652521 0.0774105 0.0386492 13 [0.225,0.237) 0.230723 0.0140705 0.0177333 0.0369638 0.0672822 0.0764555 0.0402684 14 [0.237,0.249) 0.242818 0.0146810 0.0186397 0.0381558 0.0692375 0.0786225 0.0413928 15 [0.249, oo) 0.279499 0.0168309 0.0207854 0.0401272 0.0721349 0.0829652 0.0439267 Advances in Operations Research 17 𝑖 𝑖 𝑖 𝑖 Table 9: Continued. p–quantiles of MAEs among w and w∗(LLSM) i VRCM for TSL(A)Mean TSL(A)inVRCM Average MAEs among w and w ∗ (LLSM) p=0.05 p=0.1 p=0.5 p=0.9 p=0.95 1 [0,0.07975) 0.061626 0.00329141 0.0040063 0.0079292 0.0153184 0.017980 0.0089902 2 [0.07975,0.09) 0.085354 0.00558449 0.0066781 0.0124836 0.0217916 0.0254877 0.0136394 3 [0.09,0.10) 0.095128 0.00622084 0.0074651 0.0136346 0.0241580 0.0288301 0.0151410 4 [0.10,0.11) 0.105046 0.00677146 0.0081844 0.0150571 0.0277432 0.0343089 0.0170348 5 [0.11,0.12) 0.114884 0.00728075 0.0089529 0.0164708 0.0329745 0.0408782 0.0192642 6 [0.12,0.13) 0.124902 0.00792417 0.0097471 0.0185168 0.0378364 0.0464765 0.0217170 7 [0.13,0.14) 0.134949 0.00851189 0.0104389 0.0202075 0.0415434 0.0507830 0.0236614 8 [0.14,0.15) 0.144883 0.00952136 0.0115606 0.0224145 0.0446314 0.0531641 0.0257116 9 [0.15,0.161) 0.155416 0.0101888 0.0121602 0.0241178 0.0465694 0.0553538 0.0275101 10 [0.161,0.171) 0.165845 0.0110535 0.0132394 0.0261677 0.0499157 0.0583309 0.0293786 11 [0.171,0.181) 0.175874 0.0116123 0.0139639 0.0273006 0.0515428 0.0596329 0.0304575 12 [0.181,0.191) 0.185981 0.0121824 0.0150547 0.0293308 0.0532065 0.0613544 0.0320030 13 [0.191,0.201) 0.195819 0.0122294 0.0152015 0.0299135 0.0553010 0.0642011 0.0330142 14 [0.201,0.211) 0.205937 0.0132402 0.0164008 0.0321805 0.0552598 0.0636846 0.0343310 15 [0.211, oo) 0.235348 0.0147413 0.0179580 0.0321805 0.0586515 0.0682411 0.0363445 p–quantiles of MAEs among w and w∗(LLSM) i VRCM for TSL(A)Mean TSL(A)inVRCM Average MAEs among w and w ∗ (LLSM) p=0.05 p=0.1 p=0.5 p=0.9 p=0.95 1 [0,0.06861) 0.056493 0.0029930 0.0036359 0.0071723 0.0129253 0.0151100 0.0079193 2 [0.06861,0.078) 0.073616 0.0047668 0.0056647 0.0098201 0.0165545 0.0197820 0.0107659 3 [0.078,0.087) 0.082558 0.0051148 0.00615425 0.0108293 0.0189720 0.0232764 0.0121106 4 [0.087,0.095) 0.090957 0.0054815 0.0067644 0.0120701 0.0230822 0.0289636 0.0139455 5 [0.095,0.104) 0.0995085 0.0062208 0.0074045 0.0134360 0.0267488 0.0338094 0.0157422 6 [0.104,0.113) 0.108507 0.0065308 0.0079148 0.0148310 0.0307300 0.0379708 0.0175495 7 [0.113,0.122) 0.117503 0.0073636 0.0087983 0.0166204 0.0342287 0.0402093 0.0192815 8 [0.122,0.131) 0.126447 0.0077367 0.00920785 0.0182781 0.0366835 0.0432579 0.0209778 9 [0.131,0.140) 0.135467 0.0081883 0.0099817 0.0200944 0.0391024 0.0463982 0.0227669 10 [0.140,0.149) 0.144406 0.00893715 0.0109052 0.0215995 0.0404294 0.0465999 0.0240918 11 [0.149,0.158) 0.153395 0.0096365 0.0118788 0.0228543 0.0420224 0.0488816 0.0252208 12 [0.158,0.167) 0.162379 0.0105213 0.0128739 0.0250637 0.0441591 0.0509963 0.0270496 13 [0.167,0.176) 0.171319 0.0109917 0.0133182 0.0253654 0.0446033 0.0525163 0.0275815 14 [0.176,0.185) 0.180246 0.0120041 0.0144395 0.0266159 0.0464516 0.0529339 0.0289197 15 [0.185, oo) 0.205854 0.0127740 0.0155662 0.0283804 0.0479564 0.0549310 0.0304352 18 Advances in Operations Research Table 9: Continued. p–quantiles of MAEs among w and w∗(LLSM) i VRCM for TSL(A)Mean TSL(A)inVRCM Average MAEs among w and w ∗ (LLSM) p=0.05 p=0.1 p=0.5 p=0.9 p=0.95 1 [0,0.059795) 0.051197 0.0026372 0.0031677 0.0061278 0.0107141 0.0122502 0.0066588 2 [0.05979,0.068) 0.064092 0.0040166 0.0047872 0.0079722 0.0133870 0.0158901 0.0087678 3 [0.068,0.076) 0.072019 0.0044127 0.0052033 0.0089180 0.0154595 0.0189767 0.0099818 4 [0.076,0.085) 0.080495 0.0046625 0.0055923 0.0098746 0.0183837 0.0234965 0.0114040 5 [0.085,0.093) 0.089047 0.0052813 0.0062378 0.0110158 0.0221043 0.0279174 0.0129826 6 [0.093,0.101) 0.097017 0.0056575 0.0067669 0.0124636 0.0261396 0.0326188 0.0147615 7 [0.101,0.109) 0.105051 0.0061505 0.0074036 0.0138920 0.0290254 0.0358010 0.0164984 8 [0.109,0.118) 0.113488 0.0066692 0.0079686 0.0153474 0.0308319 0.0365922 0.0177312 9 [0.118,0.126) 0.122009 0.0073133 0.0087907 0.0171076 0.0330852 0.0388189 0.0193438 10 [0.126,0.134) 0.129857 0.0076181 0.0092912 0.0186416 0.0343317 0.0401595 0.0204982 11 [0.134,0.142) 0.137970 0.0083801 0.0102174 0.0199779 0.0355818 0.0416818 0.0216939 12 [0.142,0.151) 0.146298 0.0091112 0.0107807 0.0212040 0.0376109 0.0430078 0.0229256 13 [0.151,0.159) 0.154883 0.0097330 0.0118942 0.0219635 0.0378245 0.0435528 0.0237785 14 [0.159,0.167) 0.162793 0.0102563 0.0125995 0.0228089 0.0390630 0.0443591 0.0244409 15 [0.167, oo) 0.184864 0.0115601 0.0138072 0.0242891 0.0403012 0.0468790 0.0259996 Advances in Operations Research 19 𝑖 𝑖 𝑖 𝑖 Th Table 10: Performance of TSL(A) index under the action of LLSM as the PP. Statistical characteristics of the MREs distribution in relation to various levels of TSL(A)withinagivenVRCM for i=1,...,15. e results are based on 10,000 perturbed random reciprocal PCMs with application of geometric scale and were generated on the basis of SA|K| as the simulation algorithm. The table contains results for ∈ {5,6,7,8,9} , presented consecutively. p–quantiles of MREs among w and w∗(LLSM) i VRCM for TSL(A)Mean TSL(A)in VRCM Average MREs among w and w ∗ (LLSM) p=0.05 p=0.1 p=0.5 p=0.9 p=0.95 1 [0,0.0899) 0.057816 0.025386 0.031105 0.063148 0.125229 0.157531 0.078369 2 [0.0899,0.107) 0.099189 0.037543 0.046524 0.094170 0.210311 0.285465 0.134037 3 [0.107,0.124) 0.116009 0.041496 0.051655 0.111830 0.248076 0.333749 0.162180 4 [0.124,0.141) 0.133320 0.052169 0.066257 0.141688 0.280316 0.388208 0.192453 5 [0.141,0.159) 0.150737 0.069557 0.088451 0.153061 0.322629 0.435671 0.228918 6 [0.159,0.176) 0.167818 0.083038 0.099089 0.176675 0.380074 0.531411 0.254632 7 [0.176,0.193) 0.184601 0.089335 0.106115 0.189599 0.418380 0.584780 0.282962 8 [0.193,0.21) 0.201376 0.094216 0.114126 0.200407 0.457435 0.650873 0.321145 9 [0.21,0.227) 0.218386 0.100531 0.122845 0.211782 0.509984 0.738768 0.357627 10 [0.227,0.244) 0.235167 0.104517 0.126347 0.229201 0.550244 0.802552 0.405093 11 [0.244,0.262) 0.252630 0.110690 0.133916 0.259418 0.623605 0.914232 0.454293 12 [0.262,0.279) 0.270118 0.114260 0.139642 0.275259 0.633504 0.926234 0.472063 13 [0.279,0.296) 0.287110 0.119073 0.144920 0.285331 0.656276 0.937927 0.463537 14 [0.296,0.313) 0.304180 0.124614 0.150476 0.293591 0.674574 0.982104 0.481352 15 [0.313, oo) 0.355480 0.130391 0.159726 0.311994 0.714442 1.105550 0.530692 p–quantiles of MREs among w and w∗(LLSM) i VRCM for TSL(A)Mean TSL(A)in VRCM Average MREs among w and w ∗ (LLSM) p=0.05 p=0.1 p=0.5 p=0.9 p=0.95 1 [0,0.0905) 0.063152 0.026955 0.032403 0.061404 0.122756 0.150047 0.091929 2 [0.0905,0.103) 0.097447 0.047882 0.058260 0.106461 0.193601 0.263458 0.169359 3 [0.103,0.115) 0.109447 0.055524 0.068321 0.119658 0.237508 0.340020 0.188758 4 [0.115,0.127) 0.121265 0.065842 0.078555 0.135479 0.288241 0.416876 0.228675 5 [0.127,0.14) 0.133603 0.071783 0.085975 0.147552 0.336650 0.492428 0.236603 6 [0.14,0.152) 0.145988 0.078850 0.093398 0.160596 0.399013 0.598307 0.288555 7 [0.152,0.164) 0.157828 0.083621 0.100037 0.181155 0.460463 0.713230 0.328321 8 [0.164,0.176) 0.169764 0.088879 0.106790 0.204747 0.509118 0.799108 0.392186 9 [0.176,0.189) 0.182294 0.095569 0.114805 0.224679 0.539696 0.816097 0.387082 10 [0.189,0.201) 0.194851 0.101796 0.122377 0.241424 0.569156 0.840488 0.403202 11 [0.201,0.213) 0.206810 0.106787 0.130048 0.253098 0.584213 0.886693 0.414521 12 [0.213,0.225) 0.218759 0.111142 0.135331 0.263625 0.597492 0.909918 0.427723 13 [0.225,0.238) 0.231244 0.115692 0.142985 0.274482 0.629127 1.011140 0.458373 14 [0.238,0.25) 0.243689 0.119538 0.145306 0.282161 0.629213 1.016580 0.492177 15 [0.25, oo) 0.281051 0.131983 0.160474 0.301961 0.725361 1.305030 0.569886 20 Advances in Operations Research 𝑖 𝑖 𝑖 𝑖 Table 10: Continued. p–quantiles of MREs among w and w∗(LLSM) i VRCM for TSL(A)Mean TSL(A)inVRCM Average MREs among w and w ∗ (LLSM) p=0.05 p=0.1 p=0.5 p=0.9 p=0.95 1 [0,0.07935) 0.061389 0.027453 0.033069 0.064786 0.121032 0.148877 0.091901 2 [0.07935,0.089) 0.084618 0.048742 0.058162 0.097221 0.180521 0.258818 0.150911 3 [0.089,0.1) 0.094726 0.055028 0.064762 0.108760 0.227929 0.354122 0.188055 4 [0.1,0.11) 0.105058 0.060689 0.072055 0.119836 0.288762 0.451518 0.223222 5 [0.11,0.12) 0.114936 0.066323 0.077914 0.135084 0.360043 0.586682 0.258300 6 [0.12,0.13) 0.124948 0.071119 0.084561 0.156093 0.418538 0.673318 0.310323 7 [0.13,0.14) 0.134942 0.076207 0.091276 0.176892 0.459282 0.723317 0.339038 8 [0.14,0.15) 0.144913 0.082067 0.099388 0.197851 0.491089 0.760550 0.360014 9 [0.15,0.16) 0.154903 0.089571 0.107351 0.214104 0.510660 0.781063 0.370044 10 [0.16,0.17) 0.164889 0.094907 0.113971 0.227829 0.527074 0.821481 0.390846 11 [0.17,0.181) 0.175348 0.103233 0.124254 0.244012 0.543260 0.818923 0.403747 12 [0.181,0.191) 0.185853 0.109529 0.131299 0.251883 0.574974 0.893017 0.417672 13 [0.191,0.201) 0.195779 0.115457 0.138375 0.261156 0.587231 0.940269 0.442622 14 [0.201,0.211) 0.205771 0.122449 0.144794 0.267270 0.591900 0.908632 0.452321 15 [0.211, oo) 0.235657 0.132645 0.158647 0.289569 0.697684 1.246370 0.540476 p–quantiles of MREs among w and w∗(LLSM) i VRCM for TSL(A)Mean TSL(A)inVRCM Average MREs among w and w ∗ (LLSM) p=0.05 p=0.1 p=0.5 p=0.9 p=0.95 1 [0,0.0687) 0.056815 0.028298 0.033736 0.064678 0.120167 0.158884 0.090449 2 [0.0687,0.078) 0.073695 0.047574 0.055250 0.089089 0.176783 0.305679 0.164938 3 [0.078,0.087) 0.082568 0.050834 0.059422 0.099121 0.231914 0.395970 0.195522 4 [0.087,0.096) 0.091492 0.055367 0.065374 0.113529 0.317872 0.516773 0.237766 5 [0.096,0.105) 0.100445 0.060858 0.072058 0.129920 0.373219 0.601663 0.264844 6 [0.105,0.114) 0.109477 0.066252 0.078744 0.146265 0.410744 0.653367 0.295777 7 [0.114,0.123) 0.118453 0.072271 0.085780 0.167583 0.439425 0.692378 0.307610 8 [0.123,0.132) 0.127465 0.077733 0.092503 0.185631 0.463697 0.714777 0.319999 9 [0.132,0.141) 0.136439 0.083953 0.100942 0.204941 0.498413 0.736361 0.332616 10 [0.141,0.15) 0.145370 0.090884 0.108899 0.219291 0.513393 0.782983 0.349392 11 [0.15,0.159) 0.154363 0.098619 0.118239 0.232536 0.541734 0.833077 0.377437 12 [0.159,0.168) 0.163371 0.104881 0.126765 0.240974 0.550401 0.870980 0.403649 13 [0.168,0.177) 0.172341 0.111783 0.134349 0.252118 0.561672 0.855287 0.415888 14 [0.177,0.186) 0.181318 0.117385 0.140871 0.257608 0.593642 0.961143 0.434641 15 [0.186, oo) 0.206773 0.130867 0.155858 0.280862 0.679144 1.287050 0.519607 Advances in Operations Research 21 Table 10: Continued. p–quantiles of MREs among w and w∗(LLSM) i VRCM for TSL(A)Mean TSL(A)in VRCM Average MREs among w and w ∗ (LLSM) p=0.05 p=0.1 p=0.5 p=0.9 p=0.95 1 [0,0.0599) 0.051235 0.028109 0.032911 0.061763 0.117622 0.182188 0.097775 2 [0.0599,0.068) 0.064180 0.044045 0.050531 0.079861 0.180490 0.351457 0.153068 3 [0.068,0.076) 0.072026 0.047540 0.054987 0.090918 0.249092 0.452849 0.184488 4 [0.076,0.085) 0.080534 0.050473 0.059381 0.104907 0.316722 0.541229 0.212364 5 [0.085,0.093) 0.089006 0.056571 0.066786 0.122096 0.368507 0.60405 0.249676 6 [0.093,0.101) 0.096996 0.061878 0.073179 0.138248 0.411438 0.664728 0.271985 7 [0.101,0.109) 0.104957 0.066931 0.079049 0.157497 0.442368 0.692257 0.283826 8 [0.109,0.118) 0.113442 0.072436 0.085955 0.178618 0.466070 0.725716 0.307943 9 [0.118,0.126) 0.121941 0.077338 0.093199 0.196075 0.486246 0.762465 0.324215 10 [0.126,0.134) 0.129922 0.084719 0.101779 0.210156 0.505687 0.795012 0.344586 11 [0.134,0.142) 0.137902 0.091598 0.110772 0.222684 0.521883 0.834840 0.354677 12 [0.142,0.151) 0.146341 0.098879 0.120136 0.232111 0.531432 0.860653 0.375085 13 [0.151,0.159) 0.154800 0.107414 0.129619 0.241998 0.548790 0.897229 0.385119 14 [0.159,0.167) 0.162810 0.113641 0.136610 0.248387 0.574881 0.955015 0.375036 15 [0.167, oo) 0.185022 0.128570 0.152627 0.272902 0.706691 1.342150 0.542536 22 Advances in Operations Research popular PP, i.e., LLSM, preference scales, and number of [6] L. 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An Examination of Ranking Quality for Simulated Pairwise Judgments in relation to Performance of the Selected Consistency Measure

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Copyright © 2019 Paul Thaddeus Kazibudzki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Advances in Operations Research Volume 2019, Article ID 3574263, 24 pages https://doi.org/10.1155/2019/3574263 Research Article An Examination of Ranking Quality for Simulated Pairwise Judgments in relation to Performance of the Selected Consistency Measure Paul Thaddeus Kazibudzki Universite Internationale Jean-Paul II de Bafang, B.P. 213, Bafang, Cameroon Correspondence should be addressed to Paul aTh ddeus Kazibudzki; emailpoczta@gmail.com Received 16 October 2018; Revised 7 December 2018; Accepted 24 December 2018; Published 3 February 2019 Academic Editor: Eduardo Fernandez Copyright © 2019 Paul Thaddeus Kazibudzki. is Th is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An overview of current debates and contemporary research devoted to modeling decision making processes and their facilitation directs attention to techniques based on pairwise judgments. At the core of these techniques are various judgment consistency measures which, in a sense, control the prioritization process which leads to the establishment of decision makers’ unknown preferences. If judgments expressed by decision makers were perfectly consistent (cardinally transitive), all available prioritization techniques would deliver the same solution. However, human judgments are consistently inconsistent, as it were; thus the preference estimation quality significantly varies. The scale of these variat ions depends, among others, on the chosen consistency measure of pairwise judgments. That is why it seems important to examine relations among various consistency measures and the preferences estimation quality. This research reveals that there are consis tency measures whose performance may confuse decision makers with the quality of their ranking outcome. u Th s, it introduces a measure which is directly related to the quality of the preferences estimation process. The main problem of the research is studied via Monte Carlo simulations executed in Wolfram Mathematica Sow ft are. The research results argue that although the performance of examined consistency measures deviates from the exemplary ones in relation to the estimation quality of decision makers preferences, solutions proposed in this paper can significantly improve that quality. phenomena and relating them by making comparisons. The 1. Introduction latter method leads directly to the essence of the matter, Overwhelming scientific evidence indicates that the unaided i.e., judgments regarding a phenomenon. Judgments can be human brain is simply not capable of simultaneous analysis of relative or absolute. An absolute judgment is the relation many dieff rent, competing factors and then synthesizing the between a single stimulus and some information held in short results for the purpose of making a rational decision. Indeed, or long term memory. A relative judgment, on the other hand numerous psychological experiments, e.g., [1], including the [3], is defined as identification of some relation between two well-known Miller [2] study put forth the notion that humans stimuli both present to the observer. Certainly humans can are not capable of dealing accurately with more than about make much better relative than absolute judgments. u Th s, a pairwise comparisons method was proposed in order to seven (±2) things at a time (the human brain is limited in its short term memory capacity, its discrimination ability, and its facilitate the process of relative judgments. range of perception). Some authors proclaim [4, 5] that this method dates back to the beginning of the 20th century and was firstly applied Humans learn about anything by two means: the rfi st involves examining and studying some phenomenon from by u Th rstone [6]; however, its rs fi t scienticfi applications can the perspective of its various properties and then synthe- be found in Fechner [7]. In reality, the method itself is much sizing findings and drawing conclusions; the second entails older and its idea goes back to Ramon Lull who lived in the studying some phenomenon in relation to other similar end of 13th century. It is a fact that its popularity comes from 2 Advances in Operations Research an influential paper of Marquis de Condorcet [8]; see, e.g., as “Compare – applying a given ratio scale – your feelings [9, 10], who used this method in the election process where concerning alternative 1 versus alternative 2”, do not produce voters rank candidates based on their preference. It has been accurate outcomes. us, Th A(𝑤 )isnot established but only its perfected in many papers, e.g., [4, 11–20]. estimate A(x) containing intuitive judgments, more or less Also, the method is the core of a decision making close to A(𝑤 ) in accordance with experience, skills, specific methodology—the Analytic Hierarchy Process (AHP), devel- knowledge, personal taste, and even temporary mood or oped by Saaty [21]. Although it is criticized (see [22–25, overall disposition. In such case, consistency property does 25–28]) it is also continuously perfected and as such very not hold and the relation between elements of A(x)and A(𝑤 ) oeft n validated [29] and developed from the perspective can be expressed as𝑥 =𝑒 𝑤 where𝑒 is a perturbation factor of its utility (see [5, 30–37]). u Th s, considering the stable u fl ctuating near unity. In the statistical approach, e reefl cts increase of the AHP application (see, e.g., [38–44]) as well a realization of a random variable with a given probability as its steady position among other theories of choice, e.g., distribution. [31–37, 45–47], the problematic issues of AHP should be Besides the prioritization procedure (PP) proposed by the continuously examined and questions which arise thereof creator of AHP, right principal eigenvector method (REV), should be simultaneously addressed. there are alternative PPs devised to cope with the priority The fundamental objective of this research is to determine ratios estimation problem; their demonstrative review can be the answer to the question: Does the reduction of PCM incon- found, e.g., in [47]. Many of them are optimization based sistency lead to improvement of the priority ratios estimation and seek a vector 𝑤 , as a solution of the minimization process quality? problem given by the formula {min𝐷 (𝐴 (𝑥), 𝐴 (𝑤))} subject to some assigned constraints, such as positive coefficients and normalization condition. Because the distance function 2. Background D measures an interval between matrices A(x)and A(𝑤 ), The AHP uses the hierarchical structure of the decision different den fi itions of the distance function lead to vari- problem, pairwise relative comparisons of the elements in ous prioritization concepts and prioritization results. As an the hierarchy, and a series of redundant judgments which example, eighteen PPs in [48] are described and compared enable measurement of judgment consistency. To make a for ranking purposes although some authors suggest there proposed solution possible, i.e., derive ratio scale priorities on are only fteen fi that are different. Furthermore, since the the basis of verbal judgments, a scale is utilized to evaluate publication of the above-mentioned article, a few additional the preferences for each pair of items. Probably, the most procedures have been introduced to the literature; see, e.g., known scales are Saaty’s numerical scale which comprises [43, 49–51]. Probably the most popular alternative to the REV integers, and their reciprocals, from one (equivalent to the is the Logarithmic Least Squares Method (LLSM) developed verbal judgment, “equally preferred”) to nine (equivalent to by Crawford and Williams [26, 52]. It is given by the following the verbal judgment, “extremely preferred”), and a geometric formula: scale which usually consists of the numbers computed in 𝑛 𝑛 𝑛/2 𝑤 = min∑ ∑ ln (𝑎 ) accordance with the formula𝑓(𝑛) = 𝑐 for 𝑛 ∈ {−8,...,8}∧ (1) ()𝑀 𝑖=1 𝑗=1 𝑛∈ 𝐼 where c denotes its parameter which commonly equals 2. Other arbitrarily den fi ed numerical scales are also The LLSM solution also has the following closed form and is available, e.g., composed of arbitrary integers from one to n given by the normalized products of the elements in each row: and their reciprocals. 1/𝑛 The key issue in AHP is priority ranking on the basis of (∏ 𝑎 ) 𝑗=1 𝑤 = (2) true or approximate weights, i.e., judgments. If the relative 𝑖()𝑀𝑆𝐿𝐿 1/𝑛 𝑛 𝑛 ∑ (∏ 𝑎 ) weights of a set of activities are known, they can be expressed 𝑖=1 𝑗=1 as a Pairwise Comparison Matrix (PCM): A(𝑤 )=(𝑤 /𝑤 ),i, 𝑖 𝑗 u Th s, it is also known as the geometric mean method j=1,..., n. PCM in the AHP reflects decision makers’ prefer- anditisutilized inthisresearch which strives to improve the ences (their relative judgments) about considered activities reliability of the pairwise comparisons process which is also (criteria, scenarios, players, alternatives, etc.). On the basis of the core element of AHP. A(𝑤 ),itispossibletoderivetrueweights;i.e.,decision makers priority ratios 𝑤 ,where: i=1,...,n, are selected to be positive 3. Problem and Research Methodology and normalized to unity: ∑ 𝑤 =1. For uniformity, 𝑤 is 𝑖 𝑖 referred hereaer ft to its normalized form. If the elements There are several PCM consistency measures (PCM-CMs) of a matrix A(𝑤 ) satisfy the condition 𝑤 =1/𝑤 for all i, 𝑗𝑖 provided in literature called consistency or inconsistency j=1,..., n then matrix A(𝑤 )is called reciprocal. If the elements indices (CIs). The most popular one is the PCM-CM pro- of a matrix A(𝑤 ) satisfy the condition 𝑤 𝑤 =𝑤 for all i, posed by Saaty [21]. He proposed his CI as determined by the j, k=1,..., n,and thematrix is reciprocal, then it is called formula consistent or cardinally transitive. 𝜆 −𝑛 max Certainly, in real life situations when AHP is utilized, (3) 𝑛−1 there is no A(𝑤 ) which would reflect weights given by the vector of priority ratios. As was stated earlier, the human where n indicates the number of alternatives within the mind is not a reliable measurement device. Assignments, such particular PCM and𝜆 denotes its maximal eigenvalue. The max 𝑅𝐸𝑉 𝐶𝐼 𝑖𝑗 𝑘𝑗 𝑖𝑘 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝐿𝐿𝑆 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 Advances in Operations Research 3 significant disadvantage of the PCM-CM is the fact that it – Two formulae for its measurement: 󵄨 󵄨 can operate exclusively with reciprocal PCMs. In the case of 𝑎 𝑎 󵄨 󵄨 𝑗𝑘 󵄨 󵄨 󵄨 󵄨 𝐼𝐿𝑇 (𝑎 ,𝑎 ,𝑎 )=󵄨 ln 󵄨 (8) nonreciprocal PCMs, this measure is useless (its values are 1 𝑗𝑘 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 meaningless) which in consequence seriously diminishes the 𝑎 𝑎 value of the whole approach; see, e.g., [33]. It was also recently 𝑗𝑘 𝐼𝐿𝑇 (𝑎 ,𝑎 ,𝑎 )= ln (9) 2 𝑗𝑘 found to be incorrect; see, e.g., [4, 10, 53, 54]. However, as mentioned earlier, there are a number of – One meaningful formula for PCM-CM: additional PCM-CMs. Some of them, as in the case of CI , REV originate from the PPs devised for the purpose of the priority 𝑛−2 𝑛−1 𝑛 ratios estimation process. Their distinct feature is the fact that 𝐼𝑇 (𝐴 )= ∑ ∑ ∑ {𝐿𝐼𝑇 (𝑎 ,𝑎 ,𝑎 )} (10) 𝑥 𝑛 𝑥 𝑗𝑘 ( ) all of them can operate equally efficiently in conditions where 𝑖=1 𝑗=𝑖+1 𝑘=𝑗+1 reciprocal and nonreciprocal PCMs are accepted. Probably where x denotes the formula for triad inconsistency the most known example from that set of propositions is measurement, i.e., LTI or LTI . 1 2 PCM-CM proposed by Aguaron and Moreno-Jimenez [55] It behooves us to mention that ALTI(A)can be calculated given by the following formula: on the basis of triads from the upper triangle of the given 𝑎 𝑤 PCM when it is reciprocal or all triads within the given PCM = ∑ log ( ) 𝑀 (4) when it is nonreciprocal. (𝑛−1 )(𝑛− 2 ) 𝑤 𝑖<𝑗 As was already stated earlier, the fundamental question which should be asked by researchers who deal with the Noticeably, there are a few den fi itions of PCM-CMs problem of priority ratios estimation quality in relation to a which are not connected with any PP and are devised on the PCM consistency measure is as follows: Does the reduction of basis of the PCM consistency definition. Koczkodaj’s idea [56] PCM inconsistency lead to improvement of the priority ratios attracts attention and is the rfi st to be scrutinized. Koczkodaj’s estimation process quality? PCM-CM is grounded in his concept of triad consistency. The common reason why one strives to improve the In order to clarify this, for any three distinguished deci- consistency of the PCM, when it seems unsatisfactory, is to sion alternatives A ,A ,and A , there are three meaningful 1 2 3 increase the quality of the priority ratios estimation process. priority ratios, i.e., a , a ,and a , which have their different 𝑗𝑘 However, the above question remains open and the answer locations in a particular 𝐴 =[𝑎 ] .For some dieff rent i≤n, 𝑛×𝑛 to it is not evident. Even the creator of AHP stated once j≤n, and k≤n, the tuple (a , a , a )is called a triad. If the 𝑗𝑘 that improving consistency does not mean getting an answer matrix 𝐴 =[𝑎 ] is consistent, then a a = a for all 𝑛×𝑛 𝑗𝑘 closer to the “real” life solution [21]. It can be illustrated in the triads. following example. In consequence, either of the equations 1− 𝑎 /𝑎 𝑎 = 𝑗𝑘 Considered is the true PV (denoting true weights of 0 and 1−𝑎 𝑎 /𝑎 =0 have to be true. Taking the above 𝑗𝑘 examined alternatives), i.e., 𝑤 =[7/20, 1/4, 1/4, 3/20] and A(𝑤 ) into consideration, Koczkodaj proposed his measure for triad derived from that PV, which can be presented as follows: inconsistency by the following formula: 7 7 7 󵄨 󵄨 1 󵄨 󵄨 󵄨 󵄨 𝑎 𝑎 [ ] 󵄨 󵄨 󵄨 󵄨 𝑎 𝑗𝑘 5 5 3 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 [ ] (𝑎 ,𝑎 ,𝑎 )= min { 1− , 1− } (5) 󵄨 󵄨 󵄨 󵄨 𝑗𝑘 󵄨 󵄨 󵄨 󵄨 [ ] 󵄨 󵄨 󵄨 󵄨 𝑎 𝑎 𝑎 5 5 󵄨 󵄨 󵄨 󵄨 [ ] 𝑗𝑘 󵄨 󵄨 [ ] 7 3 [ ] 𝐴 (𝑤 )= (11) [ ] Following his idea, he then proposed the following PCM-CM [ ] 5 5 [ ] of any reciprocal PCM=A: [ ] 7 3 [ ] [ ] 3 3 3 𝐾 (𝐴 )= max{𝑇𝐼 (𝑎 ,𝑎 ,𝑎 )} 𝑛 1 𝑝 𝑗𝑘 (6) 𝑝=1,..., ( ) 𝑖<𝑗<𝑘 3 [ ] 7 5 5 Then two PCMs are considered, i.e., R(x)and A(x)pro- where the maximum value for K(A) is taken from the set of duced by a hypothetical decision maker (DM). It is assumed all possible triads in the upper triangle of a given PCM. that DM is very trustworthy and is able to express judgments On the basis of Koczkodaj’s idea of triad inconsistency, very precisely at the same time being still somehow limited by Grzybowski [5] presented his PCM-CM determined by the the necessity of expressing judgments on a scale (the example following formula: utilizes Saaty’s scale). In the rfi st scenario, entries of A(𝑤 )are rounded to Saaty’s scale and the entries are made reciprocal 𝐼 (𝐴 ) (a principal condition for a PCM in the AHP) producing 󵄨 󵄨 𝑛−2 𝑛−1 𝑛 󵄨 󵄨 󵄨 󵄨 (7) 𝑎 𝑎 󵄨 󵄨 󵄨 󵄨 1 𝑎 󵄨 󵄨 󵄨 𝑗𝑘 󵄨 󵄨 󵄨 󵄨 󵄨 11 1 2 = ∑ ∑ ∑ min { 󵄨 1− 󵄨 ,󵄨 1− 󵄨 } 𝑛 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 [ ] ( ) 𝑎 𝑎 𝑎 󵄨 󵄨 󵄨 󵄨 3 𝑗𝑘 𝑖=1 𝑗=𝑖+1 𝑘=𝑗+1 󵄨 󵄨 [ ] 11 1 2 [ ] [ ] 𝑅 (𝑥 )= (12) 11 1 2] Finally, following the idea that ln(𝑎 /𝑎 𝑎 )=−ln(𝑎 𝑎 / 𝑗𝑘 𝑗𝑘 [ ] [ ] 𝑎 ), Kazibudzki [57] redefined triad inconsistency and pro- 1 1 1 posed [ 2 2 2 ] 𝑖𝑘 𝑖𝑗 𝑖𝑗 𝑖𝑘 𝑖𝑘 𝑖𝑗 𝑖𝑘 𝑖𝑗 𝐴𝑇 𝑖𝑘 𝑖𝑗 𝑖𝑘 𝑖𝑗 𝑖𝑘 𝑖𝑗 𝑇𝐼 𝑖𝑘 𝑖𝑗 𝑖𝑘 𝑖𝑗 𝑖𝑗 𝑖𝑘 𝑖𝑘 𝑖𝑗 𝑖𝑗 𝑖𝑘 𝑖𝑗 𝑖𝑗 𝑖𝑘 𝑖𝑗 𝐿𝐿𝑆 𝐶𝐼 𝑖𝑗 𝑖𝑘 𝑖𝑗 𝐴𝐿 𝑖𝑘 𝑖𝑘 𝑖𝑗 𝑖𝑗 𝑖𝑘 𝑖𝑘 𝑖𝑗 𝑖𝑗 4 Advances in Operations Research Table 1: Values of the PCM-CMs denoted as CI(PP )for R(x) and proposed characteristics of PVs estimates (∗) quality in relation to the true PV for the case. Performance measures PP Estimates CI(PP)MAE SRC REV [0.285714,0.285714,0.285714,0.142857] 0.0 0.0357143 0.8164966 LLSM [0.285714,0.285714,0.285714,0.142857] 0.0 0.0357143 0.8164966 (∗): derived from R(x) with application of a particular PP. Table 2: Values of the PCM-CMs denoted as CI(PP )for A(x) and proposed characteristics of PVs estimates (∗) quality in relation to the true PV for the case. Performance measures PP Estimates CI(PP)MAE SRC REV [0.309401,0.267949,0.267949,0.154701] – 0.0893164 0.0202995 1 LLSM [0.314288,0.264284,0.264284,0.157144] 0.0400378 0.0178559 1 (∗): derived from A(x) with application of a particular PP. In the second scenario, only entries of A(𝑤 ) are rounded to is very true in the situation when the particular PCM is Saaty’s scale (nonreciprocal case) producing less consistent (on the basis of selected exemplary PCM- CMs). It remains to be mentioned that the inconsistency 11 1 2 problem (hence errors) does not exist for simplified pairwise [ ] comparisons [53]. [ ] [ ] 11 2 Taking into consideration only such a trivial example like [ ] [ 2 ] the one above, it becomes apparent that the relation between [ ] 𝐴 (𝑥 )= (13) [ ] performance of a consistency measure and the quality of [ ] 11 2 [ ] the priority ratios estimation process is of great importance. [ ] [ ] 1 1 1 That is why to examine the phenomenon further to improve the quality of the pairwise comparisons based prioritization [ ] 2 2 2 process was decided. u Th s, the simulation framework for It should be noted thatR(x)is perfectly consistent and this purpose was adopted from [5, 57] as the only way to A(x) is not. Tables 1 and 2 present selected values of the examine the said phenomena through computer simulations. PPs related PCM-CMs (that is, CI and CI )forR(x) REV LLSM The simulation algorithm SA|K| is comprised of the following and A(x)together with PVs derived from R(x)and A(x); phases. Mean Absolute Errors (MAEs), formula (14), among 𝑤∗ (PP) and 𝑤 for the case; Spearman Rank Correlation Coecffi ients Phase 1. Randomly generate a priority vector 𝑤 =[𝑤 , (SRCs) among 𝑤∗ (PP) and 𝑤 for the case. ...,𝑤 ] of assigned size [n x1] and related perfect PCM(𝑤 )=K(𝑤 ). 󵄨 󵄨 󵄨 󵄨 𝐸 (𝑤∗ ( ),𝑤 )= ∑ 𝑤 −𝑤 ∗ ( ) 󵄨 󵄨 (14) 󵄨 𝑖 𝑖 󵄨 Phase 2. Randomly select an element𝑤 for x<y ofK(𝑤 )and 𝑖=1 replace it with 𝑤 𝑒 where e is a relatively significant error, 𝐵 𝐵 Surprisingly, a very interesting phenomenon can be noted randomly drawn (uniform distribution) from the interval on the basis of information provided in Tables 1 and 2. The 𝑒 ∈ [2;4]. Errors of that magnitude are basically considered nonreciprocal version of the analyzed PCM contains nonzero “significant”; see, e.g., [50, 58]. values for the selected PCM-CMs. In cases similar to this example, the value of Saaty’s PCM-CM always becomes neg- Phase 3. For each other element 𝑤 , i<j≤n, select randomly a ative which makes it inexplicable and in consequence useless value e for the relatively small error in accordance with the under such circumstances (as already mentioned earlier). The given probability distribution 𝜋 (applied in equal proportions other exemplary measure taken into consideration is positive as gamma, log-normal, truncated normal,and uniform distri- and higher values than zero which indicates that particular bution) and replace the element 𝑤 with the element 𝑤 𝑒 PCM is inconsistent. On the basis of the same indicators in where e is randomly drawn (uniform distribution) from the the case of the reciprocal version of the analyzed PCM, its interval 𝑒 ∈ [0,5;1,5]. perfect consistency is apparent because all selected PCM- CMs in this case are equal to zero. However, the estimation Phase 4. Round all values of 𝑤 𝑒 for i<j ofK(𝑤 )to the precision measures (MAE and SRC), i.e., characteristics of nearest value of a considered scale. the particular PV estimation quality, indicate something quite opposite. Surprisingly, smaller values of MAEs are apparent, Phase 5. Replace all elements 𝑤 for i>j ofK(𝑤 )with 1/𝑤 . as are perfect correlations of ranks between estimated and Phase 6. After all replacements are done, return the value of genuine PV for nonreciprocal version of the analyzed PCM. Certainly, this conclusion concerns all analyzed PPs and it the examined index as well as the estimate of the vector 𝑤 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑥𝑦 𝑥𝑦 𝑃𝑃 𝑃𝑃 𝑀𝐴 Advances in Operations Research 5 denoted as 𝑤 ∗(PP) with application of assigned prioritiza- which would contradict such a pertinent relationship would tion procedure. Then return the mean absolute error (MAE) unequivocally lead to the conclusion that the examined PCM- and mean relative error (MRE) (formula (15)) between 𝑤 CM is not a good indicator of the priority ratios estimation and 𝑤 ∗(PP). Remember values computed in this phase as process quality and may mislead in further actions towards one record. acceptance or rejection of the derived priority ratios vector. 󵄨 󵄨 𝑛 An examination of the research problem depicts Figures 󵄨 󵄨 𝑤 −𝑤 ∗ 1 󵄨 ( )󵄨 𝑖 𝑖 󵄨 󵄨 𝐸 (𝑤∗ ( ),𝑤 )= ∑ (15) 1 and 2 which present the performance of Koczkodaj K(A) 𝑛 𝑤 𝑖=1 (Plots A–H). In each case hereafter a horizontal axis rep- resents values of examined PCM-CM, and a vertical axis Phase 7. Repeat Phases from Phase 2 to Phase 6 N times. denotes particular estimation errors. Noticeably, when the quality of priority ratios estimation Phase 8. Repeat Phases from Phase 1 to Phase 7 N times. in a pairwise comparisons based process is taken into consid- Phase 9. Save all records as one database file. eration, the presented relations indicate that the performance of selected PCM-CM varies from the assumption presented The above algorithm allows examining the performance earlier. The phenomenon was thus far only examined in [5]. of the selected CM in relation to the quality of the priority It may seem disturbing because the relation between priority ratios estimation process. Its framework resembles steps ratios estimation errors and analyzed PCM-CM indicates that scrutinized in the example provided earlier in this paper and the analyzed index may sometimes misinform DMs about was thoroughly described in [5]. u Th s, for brevity, it will not their judgments’ quality, affecting the vector of priorities be analyzed in detail herein. which best converge with the true vector. For formality, all parameters of the applied PDs, gamma, As seen similarly in the example provided earlier in this log-normal, truncated normal,and uniform,in the simulation paper (Tables 1 and 2), taking the particular index as the algorithm SA|K| are set in such a way that the expected measure of PCM consistency, especially when errors are small value EV(e )=1. The simulation begins from n=4. Simulations and inconsistency is not negligible, one can expect both, for n=3 are not interesting due to direct interrelation of i.e., the improvement of priority ratios estimation quality considered PCM consistency measures; see, e.g., [58, 59]. For (increase of the estimation accuracy) together with the the sake of objectivity, the simulation data is gathered in the increase of the particular CI (decrease of PCM consistency) following way: all values of selected consistency measures and, inversely, the deterioration of priority ratios estimation are split into 15 separate sets designated by the quantiles Q quality (decrease of the estimation accuracy) together with of order p from 1/15 to 14/15. The 15 intervals are defined the descent of the particular CI (improvement of PCM as the rfi st is from 0 to the quantile of order 1/15, i.e., consistency). VRCM =[0, 𝑄 ), where VRCM represents aValue Range This problem seems very troublesome especially when 1 1/15 of the Selected PCM Consistency Measure; the second denotes differences among derived priority ratios are insignificant. VRCM =[𝑄 , 𝑄 ), and so on... to the last one which Then, it may occur that a ranking order of alternatives in 2 1/15 2/15 starts from the quantile of order 14/15 and goes on to infinity, the estimated priority vector can drastically differ from the i.e., VRCM =[𝑄 ,∞). The following variables are true one because of estimation errors. From that perspective, 15 14/15 examined: mean VRCM ,average MAEwithin VRCM a necessity of controlling that issue seems paramount. In n n between 𝑤 and 𝑤∗ (PP), MAE quantiles of the following order to evaluate that problem more thoroughly, detailed orders, 0.05, 0.1, 0.5, 0.9, and 0.95, and relations between all statistical characteristics for the examined K(A)are provided of them. The application of the rounding procedure was also in Tables 3–6. It can be concluded that the performance of the assumed which in this research operates according to Saaty’s presented PCM-CM varies (compare, e.g., [9]). scale. Lastly, the scenario takes into account the compulsory The motivation for this research was to develop a PCM- assumption in conventional AHP applications, i.e., the PCM CM which could depict a more credible relation between reciprocity condition. The outcome of the simulation pro- the consistency of pairwise judgments and the priority ratios gram is presented for the most popular PP=LLSM, and the estimation quality, i.e., whose priority ratios estimation error, most attractive PCM-CM=K(A). It must be emphasized that reflected by SRC, would be very close or equal to 1 for all their other PPs and PCM-CMs were also examined but results with quantiles (the most desirable situation). their application will not be presented in this research paper Successfully, a solution of the problem was generated and because thisisbeyond itsscope. The resultsare based on is presented as follows. On the basis of triad inconsistency N =20, and N =500, i.e., 10,000 cases. measureintroduced in[57], thefollowing PCM-CM was 𝑛 𝑚 devised: 4. Results and Discussion (𝐴 ) 𝑛−2 𝑛−1 𝑛 2 One could assume that MAE and MRE quantiles of any ∑ ∑ ∑ ln (𝑎 /𝑎 𝑎 ) 𝑗𝑘 (16) 𝑖=1 𝑗=𝑖+1 𝑘=𝑗+1 order should monotonically increase concurrently with the = growth of the selected PCM-CM, e.g., VRCM index. The ( )(1 + max {ln (𝑎 /𝑎 𝑎 )} ) 3 𝑖<𝑗<𝑘 𝑝 𝑗𝑘 𝑝=1,..., ( ) same relation should occur for mean VRCM and average MAE and MRE for VRCM . The results of the proposed sim- The proposed PCM-CM is denoted as the Triads Squared ulation framework or any other similar simulation scenario Logarithm Corrected Mean and an examination of its 𝑖𝑗 𝑖𝑘 𝑖𝑗 𝑖𝑘 𝑇𝑆𝐿 𝑖𝑗 𝑃𝑃 𝑀𝑅 𝑃𝑃 6 Advances in Operations Research 0.07 Plot |A| for n=4 Plot |B| for n=4 0.020 MSRC= 0.889286 MSRC= 0.975 0.06 Quantile(0.05) Median 0.05 0.015 0.04 0.010 0.03 0.02 0.005 0.01 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.08 Plot |C| for n=4 Plot |D| for n=4 0.15 MSRC= 0.971429 MSRC= 0.985714 Quantile(0.95) Mean 0.06 0.10 0.04 0.05 0.02 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.05 0.020 Plot |E| for n=6 Plot |F| for n=6 MSRC= 0.942857 MSRC= 0.942857 0.04 Quantile(0.05) Median 0.015 0.03 0.010 0.02 0.005 0.01 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.10 0.05 Plot |G| for n=6 Plot |H| for n=6 MSRC= 0.935714 MSRC= 0.967857 0.08 0.04 Quantile(0.95) Mean 0.06 0.03 0.02 0.04 0.01 0.02 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 1: Performance of K(A)as PCM-CM. eTh plots present the relation between mean values of K(A) withinagiveninterval (VRCM ) and values of MAEs and quantiles of a given order for MAEs distribution concerning estimated and true PV for the case. The results are 𝑛/2 generated with application of LLSM as the PP and geometric scale, i.e., 𝑓(𝑛) = 𝑐 for 𝑛 ∈ {−8,...,8} ∧ 𝑛 ∈ 𝐼 as the preference scale with its most popular parameter c=2. Plots are based on 10,000 random reciprocal PCMs for n=4 and n=6. MSRC denotes MeanSpearmanRank Correlation Coefficient between analyzed variables and describes their relation’s strength. Advances in Operations Research 7 0.35 0.12 Plot |A| for n=4 Plot |B| for n=4 0.30 MSRC= 0.832143 MSRC= 0.960714 0.10 Quantile(0.05) 0.25 Median 0.08 0.20 0.06 0.15 0.04 0.10 0.02 0.05 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.6 Plot |C| for n=4 Plot |D| for n=4 1.5 MSRC= 0.939286 MSRC= 0.953571 0.5 Quantile(0.95) Mean 0.4 1.0 0.3 0.2 0.5 0.1 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.15 0.35 Plot |E| for n=6 Plot |F| for n=6 0.30 MSRC= 0.903571 MSRC= 0.953571 Quantile(0.05) Median 0.25 0.10 0.20 0.15 0.05 0.10 0.05 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.7 1.5 0.6 Plot |G| for n=6 Plot |H| for n=6 MSRC= 0.935714 MSRC= 0.771429 0.5 Quantile(0.95) Mean 1.0 0.4 0.3 0.5 0.2 0.1 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 2: Performance of K(A)as PCM-CM. eTh plots present the relation between mean values of K(A)within a given interval(VRCM ) and values of MREs and quantiles of a given order for MREs distribution concerning estimated and true PV for the case. The results are 𝑛/2 generated with application of LLSM as the PP and geometric scale, i.e., 𝑓(𝑛) = 𝑐 for 𝑛 ∈ {−8,...,8} ∧ 𝑛 ∈ 𝐼 as the preference scale with its most popular parameter c=2. Plots are based on 10,000 random reciprocal PCMs for n=4 and n=6. MSRC denotes MeanSpearmanRank Correlation Coefficient between analyzed variables and describes their relation’s strength. 8 Advances in Operations Research 𝑖 𝑝 Table 3: Performance of the K(A)index. Statistical characteristics of the MAEs distribution in relation to various VRCM for i=1,...,15 of K(A) values. The results were generated for n=4 on the basis of SA|K| as the simulation algorithm and are based on 10,000 perturbed random reciprocal PCMs. The scenario assumed LLSM as the PP and Saaty’s preference scale. –quantiles of MAEs among and 𝑤∗ (LLSM) VRCM for 𝐾() Mean 𝐾() in VRCM Average MAEs among and𝑤∗ (LLSM) = 0.05 = 0.1 = 0.5 = 0.9 = 0.95 1 [0.00,0.293) 0.123175 0.003832 0.005244 0.013085 0.031933 0.044036 0.017332 2 [0.293,0.343) 0.298548 0.004178 0.005586 0.014531 0.039371 0.057014 0.020352 3 [0.343,0.394) 0.367621 0.007113 0.009216 0.020061 0.055438 0.077366 0.027509 4 [0.394,0.444) 0.418743 0.008836 0.011492 0.024425 0.068702 0.089414 0.033247 5 [0.444,0.494) 0.468803 0.010929 0.014093 0.031328 0.080087 0.097035 0.040523 6 [0.494,0.544) 0.505368 0.006643 0.009028 0.025446 0.076131 0.099659 0.035366 7 [0.544,0.595) 0.570415 0.014658 0.019158 0.044794 0.086016 0.104479 0.049809 8 [0.595,0.645) 0.621502 0.018761 0.023914 0.047224 0.082449 0.099615 0.051410 9 [0.645,0.695) 0.660045 0.012166 0.016648 0.042915 0.089487 0.109722 0.048732 10 [0.695,0.746) 0.721332 0.020838 0.025263 0.046327 0.085889 0.101378 0.051714 11 [0.746,0.796) 0.765848 0.017353 0.022895 0.049089 0.093422 0.110678 0.054968 12 [0.796,0.846) 0.822092 0.020521 0.026287 0.054130 0.093059 0.109786 0.058010 13 [0.846,0.897) 0.871055 0.020469 0.026452 0.053883 0.095374 0.114356 0.058507 14 [0.897,0.947) 0.921011 0.019656 0.026032 0.056166 0.111384 0.132945 0.063610 15 [0.947, oo) 0.969329 0.020548 0.027742 0.070211 0.142947 0.168917 0.078853 Advances in Operations Research 9 𝑖 𝑝 Table 4: Performance of the K(A)index. Statistical characteristics of the MREs distribution in relation to various VRCM for i=1,...,15 of K(A) values. The results were generated for n=4 on the basis of SA|K| as the simulation algorithm and are based on 10,000 perturbed random reciprocal PCMs. The scenario assumed LLSM as the PP and Saaty’s preference scale. –quantiles of MREs among and𝑤∗ (LLSM) VRCM for 𝐾() Mean 𝐾() in VRCM Average MREs among and𝑤∗ (LLSM) = 0.05 = 0.1 = 0.5 = 0.9 = 0.95 1 [0.00,0.293) 0.122019 0.021006 0.028171 0.061569 0.146142 0.218195 0.096958 2 [0.293,0.343) 0.298608 0.023159 0.029979 0.071009 0.192293 0.314784 0.117319 3 [0.343,0.394) 0.367711 0.041275 0.049211 0.098279 0.296609 0.400347 0.155107 4 [0.394,0.444) 0.418546 0.049447 0.060524 0.119874 0.366732 0.524257 0.271243 5 [0.444,0.495) 0.469288 0.064006 0.075775 0.162260 0.402196 0.521439 0.282957 6 [0.495,0.545) 0.505474 0.036757 0.048691 0.127198 0.403209 0.575937 0.238749 7 [0.545,0.596) 0.571548 0.086091 0.109329 0.227951 0.454215 0.609114 0.354275 8 [0.596,0.646) 0.622419 0.123841 0.152670 0.226833 0.451058 0.627101 0.348887 9 [0.646,0.697) 0.660911 0.070502 0.096137 0.214043 0.511722 0.707955 0.349459 10 [0.697,0.747) 0.722936 0.135092 0.147302 0.217057 0.425921 0.548016 0.303217 11 [0.747,0.798) 0.767036 0.112457 0.134261 0.256527 0.514438 0.715117 0.377223 12 [0.798,0.848) 0.823550 0.122843 0.149805 0.262036 0.490951 0.662803 0.363241 13 [0.848,0.899) 0.872625 0.125868 0.158338 0.257350 0.518942 0.714696 0.377362 14 [0.899,0.949) 0.922412 0.123110 0.152628 0.284100 0.655613 0.988531 0.471624 15 [0.949, oo) 0.970250 0.123223 0.157524 0.362451 1.033090 1.822590 0.672675 10 Advances in Operations Research 𝑖 𝑝 Th Table 5: Performance of the K(A)index. Statistical characteristics of the MAEs distribution in relation to various VRCM for i=1,...,15 of K(A) values. The results were generated for n=6 on the basis of SA|K| as the simulation algorithm and are based on 10,000 perturbed random reciprocal PCMs. e scenario assumed LLSM as the PP and the geometric preference s cale. –quantiles of MREs among and𝑤∗ (LLSM) VRCM for 𝐾() Mean 𝐾() in VRCM Average MREs among and𝑤∗ (LLSM) = 0.05 = 0.1 = 0.5 = 0.9 = 0.95 1 [0.00,0.500) 0.309970 0.003138 0.003923 0.007291 0.013229 0.015591 0.008080 2 [0.500,0.537) 0.501271 0.004210 0.005171 0.010393 0.019240 0.023050 0.011567 3 [0.537,0.573) 0.554137 0.006699 0.008198 0.016071 0.029390 0.035490 0.017867 4 [0.573,0.610) 0.592713 0.007459 0.009298 0.018158 0.030929 0.035619 0.019454 5 [0.610,0.646) 0.630157 0.009699 0.011333 0.019057 0.030303 0.034814 0.020245 6 [0.646,0.683) 0.650693 0.006237 0.007816 0.015671 0.028959 0.034565 0.017455 7 [0.683,0.719) 0.702542 0.009187 0.010726 0.018481 0.030391 0.035464 0.019892 8 [0.719,0.756) 0.745895 0.008338 0.010188 0.019239 0.036328 0.043535 0.021702 9 [0.756,0.792) 0.774678 0.009249 0.010954 0.019613 0.034921 0.041489 0.021766 10 [0.792,0.829) 0.816861 0.010058 0.012214 0.023462 0.043198 0.051295 0.026023 11 [0.829,0.865) 0.847194 0.010501 0.012745 0.022585 0.038486 0.044885 0.024501 12 [0.865,0.902) 0.880103 0.011555 0.013988 0.026175 0.049286 0.058456 0.029391 13 [0.902,0.938) 0.921940 0.013008 0.016109 0.032457 0.059978 0.069519 0.035730 14 [0.938,0.975) 0.958146 0.014460 0.018227 0.037434 0.067413 0.078266 0.040733 15 [0.975, oo) 0.985451 0.019837 0.024678 0.049174 0.085952 0.097939 0.052645 Advances in Operations Research 11 Th Table 6: Performance of the K(A)index. Statistical characteristics of the MREs distribution in relation to various VRCM for i=1,...,15 of K(A) values. The results were generated for n=6 on the basis of SA|K| as the simulation algorithm and are based on 10,000 perturbed random reciprocal PCMs. e scenario assumed LLSM as the PP and the geometric preference s cale. p–quantilesof MAEsamong w and w∗(LLSM) VRCM for 𝐾() Mean 𝐾() in VRCM Average MAEs among w and w ∗ (LLSM) p=0.05 p=0.1 p=0.5 p=0.9 p=0.95 1 [0.00,0.500) 0.310089 0.023352 0.028073 0.048709 0.085736 0.100416 0.054715 2 [0.500,0.537) 0.501135 0.031293 0.037867 0.070485 0.130644 0.163143 0.146845 3 [0.537,0.573) 0.554356 0.053930 0.063208 0.116827 0.235816 0.318104 0.172693 4 [0.573,0.610) 0.592078 0.069405 0.084870 0.136276 0.226008 0.293496 0.249541 5 [0.610,0.646) 0.630342 0.087466 0.096573 0.132412 0.216063 0.282887 0.401898 6 [0.646,0.683) 0.650498 0.048560 0.059216 0.113700 0.235798 0.330528 0.226842 7 [0.683,0.719) 0.701886 0.078667 0.086119 0.125897 0.231228 0.292786 0.173261 8 [0.719,0.756) 0.746012 0.067538 0.079147 0.139081 0.326062 0.481375 0.227734 9 [0.756,0.792) 0.774741 0.073180 0.082412 0.139194 0.274548 0.366040 0.196296 10 [0.792,0.829) 0.817070 0.079380 0.094561 0.169309 0.395980 0.580617 0.273966 11 [0.829,0.865) 0.847198 0.088182 0.101460 0.153473 0.300013 0.418148 0.224248 12 [0.865,0.902) 0.880101 0.093595 0.108170 0.189377 0.449450 0.667343 0.305017 13 [0.902,0.938) 0.921891 0.105242 0.125400 0.246040 0.605379 1.034870 0.459726 14 [0.938,0.975) 0.958183 0.118728 0.145412 0.276527 0.581686 0.861373 0.433898 15 [0.975, oo) 0.985442 0.157070 0.190925 0.372889 0.888862 1.638710 0.686757 12 Advances in Operations Research 0.030 0.06 0.025 0.05 0.020 0.04 0.015 0.03 0.010 0.02 Plot |A| Plot |B| 0.005 0.01 MSRC=0.996429 MSRC=1 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.12 0.07 0.06 0.10 0.05 0.08 0.04 0.06 0.03 0.04 0.02 Plot |C| Plot |D| MSRC=1 MSRC=1 0.02 0.01 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 Figure 3: Performance of the new PCM-CM: TSL(A). eTh plots present the relation between a mean value of TSL(A)within a giveninterval (VRCM ) and quantilesof order 0.1 (Plot A), 0.5 (PlotB), and0.9 (PlotC) of MAEsdistribution aswell as MAEsaverage values (PlotD) for estimated and true PVs. The results were generated with application of LLSM as the PP and Saaty’s scale. Plots are based on 10,000 random reciprocal PCMs for n=4. performance on the basis of simulation algorithm SA|K| It is noted that all statistical characteristics of the MAEs proposed earlier in this paper was carried out (Figures 3 and and MREs distribution in relation to various VRCM for 4). i=1,...,15 of TSL(A) values, with few exceptions, monoton- As can be noticed, the proposed TSL(A)performs ically grow. This examination ascertains that the proposed credibly from the perspective of the relation among the PCM-CM is a suitable measure of relation among pairwise consistency of pairwise judgments and the priority ratios comparisons consistency and the priority ratios estimation estimation quality. It is undeniably a positive piece of infor- quality. The paramount position of the proposed TSL(A)is mation which opens a new chapter in pairwise judgments that it performs better than the other, evaluated here, PCM- based priority ratios estimation process embedded in many CM, i.e., K(A). Its position is additionally strengthened by the methodologies of decision making such as AHP. It behooves fact that its performance is similar and independent from the us to mention that TSL(A) is suitable for both reciprocal and applied PP and improves significantly for higher numbers of nonreciprocal PCMs which prospectively may improve the alternatives without regard to which PP is selected. pairwise judgments based priority ratios estimation quality It should be noted that all characteristics presented herein when nonreciprocal PCMs are accepted. are of great importance in the priority ratios estimation Tables 7 and 8 provide detailed characteristics data for process, because one has to consider the potential of rejecting TSL(A) with application of LLSM (as the most popular a “good” PCM, and vice versa, i.e., the possibility of accepting alternative to REV), Saaty’s scale, and geometric scale as the a “bad” PCM, as in the classic statistical hypothesis testing most popular preference scales. Results for other PPs are theory. However, for the rfi st time in the course of pairwise similar; thus they are not presented in order to conserve judgments based prioritization development history, the pos- the length of this paper. However, it is stressed that other sibility of selecting the level of certainty and basing decisions PPs and preference scales were also tested and examined on statistical facts has been demonstrated. during the research and their results are not depicted in For instance, considering some hypothetical PCM for this paper because they coincide with results herein pre- n=4, with its mean TSL(A)≈0.319702 for LLSM as the PP sented. (Table 7), one can expect with 95% confidence that the MAE Advances in Operations Research 13 0.15 0.30 0.25 0.10 0.20 0.15 0.05 0.10 Plot |B| Plot |A| 0.05 MSRC=0.996429 MSRC=0.978571 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.8 0.6 0.5 0.6 0.4 0.4 0.3 0.2 Plot |D| 0.2 Plot |C| 0.1 MSRC=0.996429 MSRC=1 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 Figure 4: Performance of the new PCM-CM: TSL(A). eTh plots present the relation between a mean value of TSL(A)within a giveninterval (VRCM ) and quantiles of order 0.1 (Plot A), 0.5 (Plot B), and 0.9 (Plot C) of MREs distribution as well as MREs average values (Plot D) for estimated and true PVs. eTh results were generated with application of LLSM as the PP and geometric scale. Plots are based on 10,000 random reciprocal PCMs for n=4. should not exceed the value of 0.1208420. At the same time, as previously. Assuming this time TSL(A)≈0.049049, it can be one can expect with 95% confidence that it will be higher anticipated with 95% certainty that the MAE<0.072271 which than 0.0201740 (Table 7). Whether one decides to accept rather insures that the order of alternatives ranks should such a PCM or reject it obviously depends on the quality remain unchanged. requirements of the priority ratios estimation and the attitude For similar but even more detailed calculation, MRE regarding these errors. Indeed, the outcome of the research can be applied. It is a more accurate measure of priority finally creates the potential for true consistency control in an ratios deviation; however, its straightforward application unprecedented way, i.e., directly related to the priority ratios for calculation of discrepancies within normalized priority estimation quality. vectors is problematic. For example, the following PV is considered as 𝑤 =[0.27, In order to enable other researchers to make similar 0.26,0.24,0.23] denoting DM preferences for alternatives A , analysis concerning different numbers of alternatives, the A ,A ,and A , respectively. Taking into consideration the 2 3 4 exemplary characteristics of TSL(A) performance are pro- earlier assumed level of TSL(A)≈0.319702, the order of alter- vided for n>4 in the Appendix of this article, computed with natives ranks A =1, A =2, A =3, and A =4 can be very decep- 1 2 3 4 application of commonly known PP, i.e., LLSM and the most tive and is rather meaningless. In such a situation one can common Saaty’s scale (Table 9) and geometric scale (Table 10) expect with 95% confidence that the MAE >0.0201740 which as prospective preferences scales selected by decision makers. makes one aware that the true rank order of examined prefer- ences may appear otherwise, due to estimation errors related to DM inconsistency, e.g., 𝑤∗ =[(0.27–0.025), (0.26–0.025), 5. Conclusions (0.24+0.025), (0.23+0.025)] = [0.242, 0.231, 0.269, 0.258], with MAE=0.025, which designates a different order for In this research, the performance of the selected PCM- alternatives ranks, A =3, A =4, A =1, and A =2. On the CM from the perspective of its relation between pairwise 1 2 3 4 other hand, consider PV as 𝑤 =[0.49,0.33,0.16,0.02] of DM judgments consistency and the quality of the priority ratios preferences for alternatives A ,A ,A ,and A consecutively, estimation process was examined with application of the most 1 2 3 4 14 Advances in Operations Research Table 7: Performance of the TSL(A)index. Statistical characteristics of the MAEs distribution in relation to various VRCM for i=1,...,15 of TSL(A) values. The results were generated for n=4 on the basis of SA|K| as the simulation algorithm and are based on 10,000 perturbed random reciprocal PCMs. The scenario assumed LLSM as the PP and Saaty’s preference scale. –quantiles of MAEs among and𝑤∗ (LLSM) VRCM for (𝑇𝑆 𝐴) Mean (𝑇𝑆 𝐴) in VRCM Average MAEs among and𝑤∗ (LLSM) = 0.05 = 0.1 = 0.5 = 0.9 = 0.95 1 [0.00,0.0934) 0.049049 0.005253 0.007007 0.017709 0.047828 0.072271 0.024607 2 [0.0934,0.12) 0.108368 0.006543 0.008965 0.022188 0.064636 0.089647 0.030783 3 [0.120,0.147) 0.131977 0.007560 0.010195 0.025401 0.073557 0.096264 0.034682 4 [0.147,0.173) 0.161289 0.009281 0.012408 0.032397 0.084857 0.106224 0.041842 5 [0.173,0.200) 0.186567 0.010605 0.014283 0.039235 0.087242 0.107040 0.046369 6 [0.200,0.227) 0.213651 0.012731 0.017479 0.044342 0.092117 0.112116 0.050310 7 [0.227,0.253) 0.240868 0.017165 0.022310 0.047478 0.093918 0.112928 0.053405 8 [0.253,0.280) 0.267645 0.018953 0.024106 0.048903 0.095909 0.112627 0.055422 9 [0.280,0.307) 0.293803 0.020081 0.025244 0.052348 0.097503 0.114723 0.058089 10 [0.307,0.333) 0.319702 0.020174 0.025936 0.054471 0.101461 0.120842 0.060164 11 [0.333,0.360) 0.345876 0.021179 0.027149 0.055049 0.104366 0.126714 0.061558 12 [0.360,0.387) 0.372744 0.021740 0.027979 0.056325 0.107628 0.130267 0.063053 13 [0.387,0.413) 0.400500 0.022274 0.028479 0.057966 0.110502 0.132659 0.064974 14 [0.413,0.440) 0.425325 0.021991 0.028264 0.060355 0.116391 0.137831 0.067429 15 [0.440, oo) 0.509413 0.022479 0.029361 0.063909 0.122018 0.144845 0.071126 𝐿𝐿𝑆 Advances in Operations Research 15 Th Table 8: Performance of the TSL(A)index. Statistical characteristics of the MREs distribution in relation to various VRCM for i=1,...,15 of TSL(A) values. The results were generated for n=4 on the basis of SA|K| as the simulation algorithm and are based on 10,000 perturbed random reciprocal PCMs. e scenario assumed LLSM as the PP and geometric scale for prefer ences. –quantiles of MREs among and𝑤∗ () VRCM for (𝑇𝑆 𝐴) Mean (𝑇𝑆 𝐴) in VRCM Average MREs among and𝑤∗ (LLSM) = 0.05 = 0.1 = 0.5 = 0.9 = 0.95 1 [0.00,0.063) 0.038493 0.021222 0.027957 0.063897 0.162196 0.250369 0.101603 2 [0.063,0.092) 0.075288 0.043098 0.049606 0.093880 0.275593 0.404725 0.159070 3 [0.092,0.120) 0.108927 0.030409 0.041118 0.100685 0.327138 0.441926 0.170932 4 [0.120,0.149) 0.131428 0.038427 0.049762 0.121698 0.379335 0.529179 0.199342 5 [0.149,0.177) 0.165031 0.054076 0.070318 0.180633 0.449114 0.626229 0.285351 6 [0.177,0.206) 0.194473 0.051717 0.069238 0.199856 0.462612 0.640247 0.301912 7 [0.206,0.234) 0.220825 0.089354 0.120756 0.223531 0.469640 0.630734 0.320947 8 [0.234,0.263) 0.250293 0.108335 0.138489 0.218002 0.465311 0.629102 0.324638 9 [0.263,0.291) 0.277319 0.114172 0.135641 0.246244 0.504231 0.682016 0.356697 10 [0.291,0.320) 0.306276 0.120962 0.143374 0.258090 0.507218 0.698669 0.361393 11 [0.320,0.348) 0.333741 0.122310 0.147515 0.260439 0.552231 0.775952 0.390815 12 [0.348,0.377) 0.361859 0.123757 0.156182 0.263475 0.579019 0.842248 0.425032 13 [0.377,0.405) 0.390045 0.133595 0.160033 0.267660 0.619568 0.920619 0.443646 14 [0.405,0.434) 0.416203 0.126431 0.153950 0.289740 0.699219 1.141290 0.523514 15 [0.434, oo) 0.506589 0.129256 0.162990 0.328123 0.834585 1.430670 0.632685 16 Advances in Operations Research Th 𝑖 𝑖 Table 9: Performance of TSL(A) index under the action of LLSM as the PP. Statistical characteristics of the MAEs distribution in relation to various levels of TSL(A) withina givenVRCM for i=1,...,15. The results are based on 10,000 perturbed random reciprocal PCMs with application of Saaty’s scales and were generated on the basis of SA|K| as the simulation algorithm. e table contains results for ∈ {5,6,7,8,9} , presented consecutively. –quantiles of MAEs among w and w∗(LLSM) VRCM for (𝑇𝑆 𝐴) Mean (𝑇𝑆 𝐴) in VRCM Average MAEs among w and w ∗ (LLSM) =0.05 =0.1 =0.5 =0.9 =0.95 1 [0, 0.0899) 0.057912 0.0039186 0.0049954 0.0109799 0.0221753 0.0274887 0.0127898 2 [0.0899,0.107) 0.099124 0.0056158 0.0073876 0.0157136 0.0324243 0.0398139 0.0183201 3 [0.107,0.124) 0.116088 0.0063140 0.0079525 0.0184299 0.0389673 0.0490687 0.0214159 4 [0.124,0.142) 0.133907 0.0075132 0.0102233 0.0230429 0.0443459 0.0539668 0.0258898 5 [0.142,0.159) 0.151127 0.0099921 0.0129535 0.0261046 0.0486044 0.0581258 0.0290851 6 [0.159,0.176) 0.167911 0.0113191 0.0142543 0.0289546 0.0558904 0.0682777 0.0328936 7 [0.176,0.193) 0.184671 0.0125612 0.0158052 0.0320054 0.0594491 0.0730399 0.0357402 8 [0.193,0.211) 0.201896 0.0136853 0.0171375 0.0339101 0.0640703 0.0789391 0.0380755 9 [0.211,0.228) 0.219329 0.0142803 0.0178080 0.0361548 0.0711273 0.0839402 0.0408705 10 [0.228,0.245) 0.236371 0.0150518 0.0185369 0.0380656 0.0762136 0.0919801 0.0435024 11 [0.245,0.262) 0.253302 0.0161087 0.0208189 0.0405464 0.0789105 0.0929572 0.0462684 12 [0.262,0.280) 0.270523 0.0160427 0.0205586 0.0431223 0.0821647 0.0965329 0.0482168 13 [0.280,0.297) 0.288211 0.0165698 0.0209757 0.0457022 0.0865715 0.100490 0.0504072 14 [0.297,0.314) 0.305099 0.0177870 0.0226112 0.0455671 0.0859316 0.100544 0.0507868 15 [0.314, oo) 0.357080 0.0186614 0.0241816 0.0493007 0.0932224 0.107664 0.0547348 p–quantiles of MAEs among w and w∗(LLSM) i VRCM for TSL(A)Mean TSL(A)inVRCM Average MAEs among w and w ∗ (LLSM) p=0.05 p=0.1 p=0.5 p=0.9 p=0.95 1 [0, 0.0901) 0.0618775 0.0036042 0.0044511 0.0090509 0.0175099 0.0212066 0.0102634 2 [0.0901,0.102) 0.096694 0.00584585 0.0072472 0.0147877 0.0255157 0.0297767 0.0158427 3 [0.102,0.115) 0.109186 0.0071783 0.0088408 0.0167774 0.0304119 0.0360603 0.0186253 4 [0.115,0.127) 0.121228 0.00831565 0.0102091 0.0192100 0.0349865 0.0420209 0.0214601 5 [0.127,0.139) 0.133028 0.0088771 0.0109435 0.0208206 0.0393504 0.0481357 0.0236802 6 [0.139,0.151) 0.144977 0.0097898 0.0118208 0.0225534 0.0439163 0.0538868 0.0259512 7 [0.151,0.163) 0.156874 0.0101678 0.0126009 0.0248914 0.0500528 0.0613696 0.0288113 8 [0.163,0.176) 0.169306 0.0113233 0.0138144 0.0274455 0.0552421 0.0656847 0.0317599 9 [0.176,0.188) 0.181783 0.0120341 0.0147276 0.0297646 0.0587297 0.0700824 0.0339487 10 [0.188,0.200) 0.193745 0.0124796 0.0157621 0.0317564 0.0613300 0.0720410 0.0356610 11 [0.200,0.212) 0.205758 0.0137977 0.0167981 0.0329443 0.0622977 0.0721443 0.0368687 12 [0.212,0.225) 0.218204 0.0140878 0.0175574 0.0347152 0.0652521 0.0774105 0.0386492 13 [0.225,0.237) 0.230723 0.0140705 0.0177333 0.0369638 0.0672822 0.0764555 0.0402684 14 [0.237,0.249) 0.242818 0.0146810 0.0186397 0.0381558 0.0692375 0.0786225 0.0413928 15 [0.249, oo) 0.279499 0.0168309 0.0207854 0.0401272 0.0721349 0.0829652 0.0439267 Advances in Operations Research 17 𝑖 𝑖 𝑖 𝑖 Table 9: Continued. p–quantiles of MAEs among w and w∗(LLSM) i VRCM for TSL(A)Mean TSL(A)inVRCM Average MAEs among w and w ∗ (LLSM) p=0.05 p=0.1 p=0.5 p=0.9 p=0.95 1 [0,0.07975) 0.061626 0.00329141 0.0040063 0.0079292 0.0153184 0.017980 0.0089902 2 [0.07975,0.09) 0.085354 0.00558449 0.0066781 0.0124836 0.0217916 0.0254877 0.0136394 3 [0.09,0.10) 0.095128 0.00622084 0.0074651 0.0136346 0.0241580 0.0288301 0.0151410 4 [0.10,0.11) 0.105046 0.00677146 0.0081844 0.0150571 0.0277432 0.0343089 0.0170348 5 [0.11,0.12) 0.114884 0.00728075 0.0089529 0.0164708 0.0329745 0.0408782 0.0192642 6 [0.12,0.13) 0.124902 0.00792417 0.0097471 0.0185168 0.0378364 0.0464765 0.0217170 7 [0.13,0.14) 0.134949 0.00851189 0.0104389 0.0202075 0.0415434 0.0507830 0.0236614 8 [0.14,0.15) 0.144883 0.00952136 0.0115606 0.0224145 0.0446314 0.0531641 0.0257116 9 [0.15,0.161) 0.155416 0.0101888 0.0121602 0.0241178 0.0465694 0.0553538 0.0275101 10 [0.161,0.171) 0.165845 0.0110535 0.0132394 0.0261677 0.0499157 0.0583309 0.0293786 11 [0.171,0.181) 0.175874 0.0116123 0.0139639 0.0273006 0.0515428 0.0596329 0.0304575 12 [0.181,0.191) 0.185981 0.0121824 0.0150547 0.0293308 0.0532065 0.0613544 0.0320030 13 [0.191,0.201) 0.195819 0.0122294 0.0152015 0.0299135 0.0553010 0.0642011 0.0330142 14 [0.201,0.211) 0.205937 0.0132402 0.0164008 0.0321805 0.0552598 0.0636846 0.0343310 15 [0.211, oo) 0.235348 0.0147413 0.0179580 0.0321805 0.0586515 0.0682411 0.0363445 p–quantiles of MAEs among w and w∗(LLSM) i VRCM for TSL(A)Mean TSL(A)inVRCM Average MAEs among w and w ∗ (LLSM) p=0.05 p=0.1 p=0.5 p=0.9 p=0.95 1 [0,0.06861) 0.056493 0.0029930 0.0036359 0.0071723 0.0129253 0.0151100 0.0079193 2 [0.06861,0.078) 0.073616 0.0047668 0.0056647 0.0098201 0.0165545 0.0197820 0.0107659 3 [0.078,0.087) 0.082558 0.0051148 0.00615425 0.0108293 0.0189720 0.0232764 0.0121106 4 [0.087,0.095) 0.090957 0.0054815 0.0067644 0.0120701 0.0230822 0.0289636 0.0139455 5 [0.095,0.104) 0.0995085 0.0062208 0.0074045 0.0134360 0.0267488 0.0338094 0.0157422 6 [0.104,0.113) 0.108507 0.0065308 0.0079148 0.0148310 0.0307300 0.0379708 0.0175495 7 [0.113,0.122) 0.117503 0.0073636 0.0087983 0.0166204 0.0342287 0.0402093 0.0192815 8 [0.122,0.131) 0.126447 0.0077367 0.00920785 0.0182781 0.0366835 0.0432579 0.0209778 9 [0.131,0.140) 0.135467 0.0081883 0.0099817 0.0200944 0.0391024 0.0463982 0.0227669 10 [0.140,0.149) 0.144406 0.00893715 0.0109052 0.0215995 0.0404294 0.0465999 0.0240918 11 [0.149,0.158) 0.153395 0.0096365 0.0118788 0.0228543 0.0420224 0.0488816 0.0252208 12 [0.158,0.167) 0.162379 0.0105213 0.0128739 0.0250637 0.0441591 0.0509963 0.0270496 13 [0.167,0.176) 0.171319 0.0109917 0.0133182 0.0253654 0.0446033 0.0525163 0.0275815 14 [0.176,0.185) 0.180246 0.0120041 0.0144395 0.0266159 0.0464516 0.0529339 0.0289197 15 [0.185, oo) 0.205854 0.0127740 0.0155662 0.0283804 0.0479564 0.0549310 0.0304352 18 Advances in Operations Research Table 9: Continued. p–quantiles of MAEs among w and w∗(LLSM) i VRCM for TSL(A)Mean TSL(A)inVRCM Average MAEs among w and w ∗ (LLSM) p=0.05 p=0.1 p=0.5 p=0.9 p=0.95 1 [0,0.059795) 0.051197 0.0026372 0.0031677 0.0061278 0.0107141 0.0122502 0.0066588 2 [0.05979,0.068) 0.064092 0.0040166 0.0047872 0.0079722 0.0133870 0.0158901 0.0087678 3 [0.068,0.076) 0.072019 0.0044127 0.0052033 0.0089180 0.0154595 0.0189767 0.0099818 4 [0.076,0.085) 0.080495 0.0046625 0.0055923 0.0098746 0.0183837 0.0234965 0.0114040 5 [0.085,0.093) 0.089047 0.0052813 0.0062378 0.0110158 0.0221043 0.0279174 0.0129826 6 [0.093,0.101) 0.097017 0.0056575 0.0067669 0.0124636 0.0261396 0.0326188 0.0147615 7 [0.101,0.109) 0.105051 0.0061505 0.0074036 0.0138920 0.0290254 0.0358010 0.0164984 8 [0.109,0.118) 0.113488 0.0066692 0.0079686 0.0153474 0.0308319 0.0365922 0.0177312 9 [0.118,0.126) 0.122009 0.0073133 0.0087907 0.0171076 0.0330852 0.0388189 0.0193438 10 [0.126,0.134) 0.129857 0.0076181 0.0092912 0.0186416 0.0343317 0.0401595 0.0204982 11 [0.134,0.142) 0.137970 0.0083801 0.0102174 0.0199779 0.0355818 0.0416818 0.0216939 12 [0.142,0.151) 0.146298 0.0091112 0.0107807 0.0212040 0.0376109 0.0430078 0.0229256 13 [0.151,0.159) 0.154883 0.0097330 0.0118942 0.0219635 0.0378245 0.0435528 0.0237785 14 [0.159,0.167) 0.162793 0.0102563 0.0125995 0.0228089 0.0390630 0.0443591 0.0244409 15 [0.167, oo) 0.184864 0.0115601 0.0138072 0.0242891 0.0403012 0.0468790 0.0259996 Advances in Operations Research 19 𝑖 𝑖 𝑖 𝑖 Th Table 10: Performance of TSL(A) index under the action of LLSM as the PP. Statistical characteristics of the MREs distribution in relation to various levels of TSL(A)withinagivenVRCM for i=1,...,15. e results are based on 10,000 perturbed random reciprocal PCMs with application of geometric scale and were generated on the basis of SA|K| as the simulation algorithm. The table contains results for ∈ {5,6,7,8,9} , presented consecutively. p–quantiles of MREs among w and w∗(LLSM) i VRCM for TSL(A)Mean TSL(A)in VRCM Average MREs among w and w ∗ (LLSM) p=0.05 p=0.1 p=0.5 p=0.9 p=0.95 1 [0,0.0899) 0.057816 0.025386 0.031105 0.063148 0.125229 0.157531 0.078369 2 [0.0899,0.107) 0.099189 0.037543 0.046524 0.094170 0.210311 0.285465 0.134037 3 [0.107,0.124) 0.116009 0.041496 0.051655 0.111830 0.248076 0.333749 0.162180 4 [0.124,0.141) 0.133320 0.052169 0.066257 0.141688 0.280316 0.388208 0.192453 5 [0.141,0.159) 0.150737 0.069557 0.088451 0.153061 0.322629 0.435671 0.228918 6 [0.159,0.176) 0.167818 0.083038 0.099089 0.176675 0.380074 0.531411 0.254632 7 [0.176,0.193) 0.184601 0.089335 0.106115 0.189599 0.418380 0.584780 0.282962 8 [0.193,0.21) 0.201376 0.094216 0.114126 0.200407 0.457435 0.650873 0.321145 9 [0.21,0.227) 0.218386 0.100531 0.122845 0.211782 0.509984 0.738768 0.357627 10 [0.227,0.244) 0.235167 0.104517 0.126347 0.229201 0.550244 0.802552 0.405093 11 [0.244,0.262) 0.252630 0.110690 0.133916 0.259418 0.623605 0.914232 0.454293 12 [0.262,0.279) 0.270118 0.114260 0.139642 0.275259 0.633504 0.926234 0.472063 13 [0.279,0.296) 0.287110 0.119073 0.144920 0.285331 0.656276 0.937927 0.463537 14 [0.296,0.313) 0.304180 0.124614 0.150476 0.293591 0.674574 0.982104 0.481352 15 [0.313, oo) 0.355480 0.130391 0.159726 0.311994 0.714442 1.105550 0.530692 p–quantiles of MREs among w and w∗(LLSM) i VRCM for TSL(A)Mean TSL(A)in VRCM Average MREs among w and w ∗ (LLSM) p=0.05 p=0.1 p=0.5 p=0.9 p=0.95 1 [0,0.0905) 0.063152 0.026955 0.032403 0.061404 0.122756 0.150047 0.091929 2 [0.0905,0.103) 0.097447 0.047882 0.058260 0.106461 0.193601 0.263458 0.169359 3 [0.103,0.115) 0.109447 0.055524 0.068321 0.119658 0.237508 0.340020 0.188758 4 [0.115,0.127) 0.121265 0.065842 0.078555 0.135479 0.288241 0.416876 0.228675 5 [0.127,0.14) 0.133603 0.071783 0.085975 0.147552 0.336650 0.492428 0.236603 6 [0.14,0.152) 0.145988 0.078850 0.093398 0.160596 0.399013 0.598307 0.288555 7 [0.152,0.164) 0.157828 0.083621 0.100037 0.181155 0.460463 0.713230 0.328321 8 [0.164,0.176) 0.169764 0.088879 0.106790 0.204747 0.509118 0.799108 0.392186 9 [0.176,0.189) 0.182294 0.095569 0.114805 0.224679 0.539696 0.816097 0.387082 10 [0.189,0.201) 0.194851 0.101796 0.122377 0.241424 0.569156 0.840488 0.403202 11 [0.201,0.213) 0.206810 0.106787 0.130048 0.253098 0.584213 0.886693 0.414521 12 [0.213,0.225) 0.218759 0.111142 0.135331 0.263625 0.597492 0.909918 0.427723 13 [0.225,0.238) 0.231244 0.115692 0.142985 0.274482 0.629127 1.011140 0.458373 14 [0.238,0.25) 0.243689 0.119538 0.145306 0.282161 0.629213 1.016580 0.492177 15 [0.25, oo) 0.281051 0.131983 0.160474 0.301961 0.725361 1.305030 0.569886 20 Advances in Operations Research 𝑖 𝑖 𝑖 𝑖 Table 10: Continued. p–quantiles of MREs among w and w∗(LLSM) i VRCM for TSL(A)Mean TSL(A)inVRCM Average MREs among w and w ∗ (LLSM) p=0.05 p=0.1 p=0.5 p=0.9 p=0.95 1 [0,0.07935) 0.061389 0.027453 0.033069 0.064786 0.121032 0.148877 0.091901 2 [0.07935,0.089) 0.084618 0.048742 0.058162 0.097221 0.180521 0.258818 0.150911 3 [0.089,0.1) 0.094726 0.055028 0.064762 0.108760 0.227929 0.354122 0.188055 4 [0.1,0.11) 0.105058 0.060689 0.072055 0.119836 0.288762 0.451518 0.223222 5 [0.11,0.12) 0.114936 0.066323 0.077914 0.135084 0.360043 0.586682 0.258300 6 [0.12,0.13) 0.124948 0.071119 0.084561 0.156093 0.418538 0.673318 0.310323 7 [0.13,0.14) 0.134942 0.076207 0.091276 0.176892 0.459282 0.723317 0.339038 8 [0.14,0.15) 0.144913 0.082067 0.099388 0.197851 0.491089 0.760550 0.360014 9 [0.15,0.16) 0.154903 0.089571 0.107351 0.214104 0.510660 0.781063 0.370044 10 [0.16,0.17) 0.164889 0.094907 0.113971 0.227829 0.527074 0.821481 0.390846 11 [0.17,0.181) 0.175348 0.103233 0.124254 0.244012 0.543260 0.818923 0.403747 12 [0.181,0.191) 0.185853 0.109529 0.131299 0.251883 0.574974 0.893017 0.417672 13 [0.191,0.201) 0.195779 0.115457 0.138375 0.261156 0.587231 0.940269 0.442622 14 [0.201,0.211) 0.205771 0.122449 0.144794 0.267270 0.591900 0.908632 0.452321 15 [0.211, oo) 0.235657 0.132645 0.158647 0.289569 0.697684 1.246370 0.540476 p–quantiles of MREs among w and w∗(LLSM) i VRCM for TSL(A)Mean TSL(A)inVRCM Average MREs among w and w ∗ (LLSM) p=0.05 p=0.1 p=0.5 p=0.9 p=0.95 1 [0,0.0687) 0.056815 0.028298 0.033736 0.064678 0.120167 0.158884 0.090449 2 [0.0687,0.078) 0.073695 0.047574 0.055250 0.089089 0.176783 0.305679 0.164938 3 [0.078,0.087) 0.082568 0.050834 0.059422 0.099121 0.231914 0.395970 0.195522 4 [0.087,0.096) 0.091492 0.055367 0.065374 0.113529 0.317872 0.516773 0.237766 5 [0.096,0.105) 0.100445 0.060858 0.072058 0.129920 0.373219 0.601663 0.264844 6 [0.105,0.114) 0.109477 0.066252 0.078744 0.146265 0.410744 0.653367 0.295777 7 [0.114,0.123) 0.118453 0.072271 0.085780 0.167583 0.439425 0.692378 0.307610 8 [0.123,0.132) 0.127465 0.077733 0.092503 0.185631 0.463697 0.714777 0.319999 9 [0.132,0.141) 0.136439 0.083953 0.100942 0.204941 0.498413 0.736361 0.332616 10 [0.141,0.15) 0.145370 0.090884 0.108899 0.219291 0.513393 0.782983 0.349392 11 [0.15,0.159) 0.154363 0.098619 0.118239 0.232536 0.541734 0.833077 0.377437 12 [0.159,0.168) 0.163371 0.104881 0.126765 0.240974 0.550401 0.870980 0.403649 13 [0.168,0.177) 0.172341 0.111783 0.134349 0.252118 0.561672 0.855287 0.415888 14 [0.177,0.186) 0.181318 0.117385 0.140871 0.257608 0.593642 0.961143 0.434641 15 [0.186, oo) 0.206773 0.130867 0.155858 0.280862 0.679144 1.287050 0.519607 Advances in Operations Research 21 Table 10: Continued. p–quantiles of MREs among w and w∗(LLSM) i VRCM for TSL(A)Mean TSL(A)in VRCM Average MREs among w and w ∗ (LLSM) p=0.05 p=0.1 p=0.5 p=0.9 p=0.95 1 [0,0.0599) 0.051235 0.028109 0.032911 0.061763 0.117622 0.182188 0.097775 2 [0.0599,0.068) 0.064180 0.044045 0.050531 0.079861 0.180490 0.351457 0.153068 3 [0.068,0.076) 0.072026 0.047540 0.054987 0.090918 0.249092 0.452849 0.184488 4 [0.076,0.085) 0.080534 0.050473 0.059381 0.104907 0.316722 0.541229 0.212364 5 [0.085,0.093) 0.089006 0.056571 0.066786 0.122096 0.368507 0.60405 0.249676 6 [0.093,0.101) 0.096996 0.061878 0.073179 0.138248 0.411438 0.664728 0.271985 7 [0.101,0.109) 0.104957 0.066931 0.079049 0.157497 0.442368 0.692257 0.283826 8 [0.109,0.118) 0.113442 0.072436 0.085955 0.178618 0.466070 0.725716 0.307943 9 [0.118,0.126) 0.121941 0.077338 0.093199 0.196075 0.486246 0.762465 0.324215 10 [0.126,0.134) 0.129922 0.084719 0.101779 0.210156 0.505687 0.795012 0.344586 11 [0.134,0.142) 0.137902 0.091598 0.110772 0.222684 0.521883 0.834840 0.354677 12 [0.142,0.151) 0.146341 0.098879 0.120136 0.232111 0.531432 0.860653 0.375085 13 [0.151,0.159) 0.154800 0.107414 0.129619 0.241998 0.548790 0.897229 0.385119 14 [0.159,0.167) 0.162810 0.113641 0.136610 0.248387 0.574881 0.955015 0.375036 15 [0.167, oo) 0.185022 0.128570 0.152627 0.272902 0.706691 1.342150 0.542536 22 Advances in Operations Research popular PP, i.e., LLSM, preference scales, and number of [6] L. 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