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Pairwise comparisons have been applied to several real decision making problems. As a result, this method has been recognized as an effective decision making tool by practitioners, experts, and researchers. Although methods based on pairwise comparisons are widespread, decision making problems with many alternatives and criteria may be challenging. This paper presents the results of an experiment used to verify the influence of a high number of preferences comparisons in the inconsistency of the comparisons matrix and identifies the influence of consistencies and inconsistencies in the assessment of the decision-making process. The findings indicate that it is difficult to predict the influence of inconsistencies and that the priority vector may or may not be influenced by low levels of inconsistencies, with a consistency ratio of less than 0.1. Finally, this work presents an interactive preference adjustment algorithm with the aim of reducing the number of pairwise comparisons while capturing effective information from the decision maker to approximate the results of the problem to their preferences. The presented approach ensures the consistency of a comparisons matrix and significantly reduces the time that decision makers need to devote to the pairwise comparisons process. An example application of the interactive preference adjustment algorithm is included.

Preference judgment is a key issue in multicriteria decision-making (MCDM) methods; MCDM is a generic term given to a collection of systematic approaches and methods developed to support the evaluation of alternatives in a context with many objectives and conflicting criteria [

Multicriteria methods can be classified into three groups: first, aggregation methods based on a single criterion of synthesis, whose main representatives are the multiattribute utility theory, the analytic hierarchy process (AHP), and MACBETH. The second group is outranking methods, such as the PROMETHEE and ELECTRE methods. Finally, the third group consists of the interactive methods, such as multiobjective linear programming (PLMO) [

Different types of cognitive and behavioral biases play an important role in decision-making (DM). The MCDM aims to help people make strategic decisions according to their preferences and an overarching understanding of the problem [

In additive models, inconsistency can be perceived in two distinct situations of decision maker (DM) judgments: intercriteria and intracriteria evaluations. The intercriteria evaluation involves the elicitation procedure for determining the weights of the criteria, where inconsistencies have been reported such as ratio [

A common cognitive deviation in MCDM methods occurs when there is a PC of a large number of alternatives, which requires a great deal of cognitive effort by the DM [

This paper focuses on the preference judgment of a DM, based on PCs of a qualitative criterion, to build a value function. For this purpose, an analysis was conducted based on the PC process. The AHP is one of the best known multicriteria decision aid (MCDA) approaches and is thus widely used [

In this article, we explore the influence of consistency and inconsistency in preference assessments—based on PCs—by performing an experiment with several individuals to assess their preferences based on a given situation. A high level of inconsistency has been verified within the literature (reference). Additionally, we evaluate an alternative procedure to assess the DM’s preferences. The preferences of a DM are assessed based on an interactive procedure of asking questions and adjusting preferences, thus reducing the time spent by the DM and assuming an acceptable level of possible inconsistencies.

This paper is organized as follows. The next section presents a review of the literature concerning the causes of inconsistencies in PCs, as well as several solutions. Section

Benitez et al. [

A PC uses human abilities, such as knowledge and experience, to compare alternatives and criteria in a pairwise manner and assemble a comparisons matrix [

The number of n PCs is

Consider the multicriteria problem, where the DM must make a PC alongside five criteria and ten alternatives. In such a situation, the DM will have to allow time to perform 225 evaluations (Table

Number of pairwise comparisons based on the number of criteria and alternatives.

Number of Alternatives | Number of Alternatives | |||||||||||||||||||
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Number of Criteria | | | | | | | | | | | | | | | | | | | ||

| 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | | 2 | 6 | 12 | 20 | 30 | 42 | 56 | 72 | 90 | |

| 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | | 3 | 9 | 18 | 30 | 45 | 63 | 84 | 108 | 135 | |

| 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | | 4 | 12 | 24 | 40 | 60 | 84 | 112 | 144 | 180 | |

| 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | | 5 | 15 | 30 | 50 | 75 | 105 | 140 | 180 | 225 | |

| 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | | 6 | 18 | 36 | 60 | 90 | 126 | 168 | 216 | 270 | |

| 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | | 7 | 21 | 42 | 70 | 105 | 147 | 196 | 252 | 315 | |

| 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | | 8 | 24 | 48 | 80 | 120 | 168 | 224 | 288 | 360 | |

| 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | | 9 | 27 | 54 | 90 | 135 | 189 | 252 | 324 | 405 | |

| 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | | 10 | 30 | 60 | 100 | 150 | 210 | 280 | 360 | 450 | |

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where

Alternatively, Equation (

where

Previous studies have already proposed several solutions to this problem. Weiss and Rao [

A comparisons matrix with a CR equal to zero is representative of a fully consistent DM; this matrix is known as a RTM or a consistent matrix [

The process of building a RTM dramatically reduces the number of comparisons performed by the DM, as shown in Table

The effort of DMs is greatly reduced when building a RTM, which may reflect a more careful evaluation of PCs. On the other hand, an error or very imprecise evaluation of the initial comparison can cause distortion in the decision process [

Numerous studies have examined problems in the use of the PC [

With regard to the ratio scale problem, Ishizaka et al. [

In some cases, an unlimited scale is required, especially when comparing measurable entities such as distance and temperature [

Another issue is the eigenvalue problem. Some authors agree that while the eigenvalue is used as a good approximation for consistent matrices, there are expressive results regarding the existence of better approximations, such as geometric means [

A PC matrix, refereed here as

When an n × n matrix

Let

If the matrix is consistent, then

Saaty [

Koczkodaj [

Koczkodaj’s consistency index [

Considering Saaty’s inconsistency index, some questions remain to be answered [

Considering Koczkodaj’s consistency index, an important issue seems to be the elaboration of the thresholds in higher dimensions or the replacement of the index with a refined rule of classification [

Wadjdi et al. [

Xu [

Pankratova and Nedashkovskaya [

A method for constructing consistent fuzzy preference relations from a set of n - 1 preference data was proposed by [

Voisin et al. [

Benítez et al. [

Benítez et al. [

Motivated by a situation found in a real application of AHP, Negahban [

Brunelli et al. [

Xia and Xu [

Benítez et al. [

The evaluation of consistency has also been studied using imprecise data. Thus, the theory of fuzzy numbers has been applied to MCDA methods to better interpret the judgment of DMs. The fuzzy AHP has been widely applied, and the evaluation of consistency in such situations has been the object of research. Bulut et al. [

Ramík and Korviny [

Xu and Wang [

Koczkodaj et al. [

A recent study examined the notion of generators of the PCs matrix [

We have performed an experiment of a decision-making situation among students and staffs of a university, using a procedure based on the PCs matrix. The experiment aims to identify the influence of the number of PCs on the comparisons matrix consistency, in accordance with the research conducted by Bozóki et al. [

The experiment consisted of presenting different location options for a summer holiday to DMs. DMs completed a comparisons matrix whose number of alternatives grew as the DM concluded a comparative stage. The alternatives were the cities of Rio de Janeiro, Florianopolis, Salvador, Natal, Fortaleza, Maceió, João Pessoa, Aracaju, Vitória, Recife, Porto Alegre, and Curitiba, as presented in Figure

Criteria hierarchy of the holiday destination choice problem.

While the experiment surveyed a sample of 180 people, only 76 answered it completely. Of the 76 who responded in full, only 30 dedicated the answers. The others only completed the questionnaire without any criteria or attention and were very inconsistent, even in the initial stages when the number of alternatives and the cognitive effort were minimal.

The evaluation process began with the comparison of three possible alternatives (cities). Then, 4 alternatives, 5 alternatives, and eventually up to 10 alternatives were evaluated. Individuals reported that, as the number of alternatives increased, their ability to discern the difference between them decreased. Lastly, the surveyed individuals were asked whether they agreed with the ranking obtained; 67% of the individuals reported that they had a different perspective than the presented results.

The results of the experiment indicate that inconsistency in the comparisons matrix increases as the number of alternatives increases, that is, as the comparisons increase (see Figure

Behavior of the consistency ratio in relation to the number of alternatives.

The findings indicate that comparisons of 3, 4, and 5 alternatives do not usually generate problems of consistency when considering CR <0.1, as recommended by Saaty, when using PCs. However, matrices with 6 or more alternatives usually generate CR> 0.1, thus extrapolating the limit.

The results further confirm the research by Bozóki et al. [

The results from the previous experiment were used to explore the influence of inconsistency on the PCs Matrix. We selected 10 alternatives (cities), in which we have observed a high rate of inconsistency in all results. The matrix values were randomly changed so as not to reverse the DM’s preferences. We analyzed the inconsistency in each example and then reproduced similar situations of inconsistency for a selected DM.

For example, if the DM compares A_{1} and A_{2}, such as 4, and A_{2} and A_{3}, as 2, to maintain consistency and transitivity, the comparison between A_{1} and A_{3} will be 6. The inconsistency introduced does not change the order A_{1} > A_{2} > A_{3}, however, although it would change the comparison between A_{1} and A_{3} to 5 or 7.

To verify the influence of inconsistencies, we created a consistent matrix formed by the RTM to serve as a reference point for each decision-maker. Therefore, a consistent comparisons matrix was created using selected responses provided by the decision-makers, i.e., the minimal necessary information. By including the additional information from the PC that was already provided by the decision-maker, we were able to generate different levels of inconsistency.

The results of a consistent matrix were compared with five other situations of inconsistency, all within an accepted level. In all situations, the matrices of comparison maintained the first line of evaluation completed by the DMs but, in each case, allowed for a different level of inconsistency (from 0 to 0.1).

The final ranking of the situations was analyzed for a given decision-maker. The results are presented in Table

Results of all evaluations of alternatives.

Cities | CR =0/Rank | CR=0.02/Rank | CR=0.04/Rank | CR=0.06/Rank | CR=0.08/Rank | CR=0.10/Rank | ||||||
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| 0.263 | 1 | 0.260 | 1 | 0.257 | 1 | 0.253 | 1 | 0.249 | 1 | 0.251 | 1 |

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| 0.052 | 6 | 0.051 | 6 | 0.051 | 6 | 0.050 | 6 | 0.051 | 6 | 0.057 | 6 |

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| 0.131 | 3 | 0.139 | 3 | 0.138 | 3 | 0.137 | 3 | 0.136 | 3 | 0.137 | 3 |

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| 0.037 | 8 | 0.037 | 8 | 0.036 | 8 | 0.036 | 9 | 0.038 | 10 | 0.046 | 8 |

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| 0.065 | 5 | 0.064 | 5 | 0.063 | 5 | 0.063 | 5 | 0.062 | 5 | 0.063 | 5 |

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| 0.258 | 2 | 0.252 | 2 | 0.249 | 2 | 0.248 | 2 | 0.249 | 1 | 0.228 | 2 |

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| 0.087 | 4 | 0.091 | 4 | 0.090 | 4 | 0.089 | 4 | 0.089 | 4 | 0.089 | 4 |

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| 0.033 | 9 | 0.035 | 9 | 0.035 | 9 | 0.034 | 10 | 0.040 | 8 | 0.040 | 9 |

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| 0.044 | 7 | 0.043 | 7 | 0.048 | 7 | 0.048 | 7 | 0.047 | 7 | 0.050 | 7 |

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| 0.029 | 10 | 0.028 | 10 | 0.032 | 10 | 0.039 | 8 | 0.038 | 9 | 0.038 | 10 |

A straightforward verification of the alternatives position in Table

A similar result has been observed for all DMs who participated in the present experiment. In all cases, at least rank changed positions. This result leads to the conclusion that we cannot rely only on CR to measure the consistency of the comparisons matrix.

Inconsistency can affect the recommendations made to the DM and can even be detrimental, resulting in a wrong decision. For example, in all ranks, Rio de Janeiro was ranked first; however, when CR = 0.08, Rio de Janeiro was tied with Fortaleza. This tie could generate doubt in the DM and cause him to choose Fortaleza. In this example a wrong decision will most likely not have detrimental consequences, but for strategic decision-making an error could provoke serious consequences.

All research related to the consistency of a comparisons matrix stems from the impossibility of ensuring that DMs are consistent with their value judgment when making PCs of the alternatives.

In the previous section, we attempted to show how the inconsistency of DMs increases as the number of alternatives increases, and how this inconsistency influences the overall results. Additionally, inconsistency is influenced by the bias in the process of DM’s elicitation of preferences.

Thus, we suggest a method that seeks to reduce and correct inconsistencies caused by DMs in the qualitative assessment of PCs.

This method is based on two procedures: First, the DM must access the preferences of a set of alternatives for a given criterion by comparing the set of alternatives to a reference alternative. The compassion process only occurs between the reference alternative, for instance, the best alternative, and the set of available alternatives. Thus, only one line of the comparisons matrix needs to be completed. In a second step, the remaining lines will be filled in with the help of the mathematical assumptions of the RTM. This ensures the consistency of the array and reduces the number of comparisons. The procedure uses the bases of the mathematical presuppositions of the RTM to verify acceptable values of inconsistencies. Later, the interactive algorithm will identify and correct the inconsistencies of the DM.

At this point, we assume that the restructuring process has already been undertaken and that the criteria and alternatives have already been identified. Decision makers subsequently sort alternatives from better to worse.

The proposed procedure reduces the number of preference comparisons demanded by the DM, while increasing the accuracy of their preferences. The DM fills in only the first row of the comparison matrices and a few pairs of alternatives of the other lines. This procedure results in a significant reduction of the number of PCs.

The interactive algorithm selects a few pairs of available alternatives to be evaluated, i.e., additional comparisons between the alternatives, to confirm the preference judgment of the DM with an estimated value in situations without inconsistency.

When DMs complete one row of a comparisons matrix, there are no inconsistencies caused by a lack of reciprocity; only deviations from judgment can occur. Nevertheless, the DM may not have been consistent with his or her own preferences. Thus, a preference adjustment in the comparisons matrix may be performed to avoid possible deviations. One way to accomplish this is proposed below.

The preference adjustment is based on questions posed to the DM. These questions review a comparison between two alternatives. These two alternatives should be chosen according to the number of alternatives.

The number of PCs has been presented previously to ensure that the minimum number of required assessments is performed. Only then can there be a good preference adjustment. Indeed, complete consistency can only be ensured if all comparison indices are evaluated. However, evaluating all comparison indices may be too expensive, thereby diminishing the advantage of reducing the number of PCs.

The number of revisions required depends on the number of alternatives (Table

For a comparisons matrix with an even number of elements n: Do not conduct assessments with comparison indices between the first row and the first column; there will be an odd number of elements to evaluate. It is not possible to form pairs with an odd number of elements without repetition, so n/2 + 1 evaluations would be needed.

For a comparisons matrix with an odd number of elements n: Again, do not conduct assessments with comparison indices between the first row and the first column; there will be an even number of elements to evaluate. In this case, n/2 evaluations would be needed.

Representation of the comparison indices that will be evaluated.

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| Comparison indices | Alternatives | Comparison indices |

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_{ 2, } _{ 3 } | | A_{2}, A_{3} | |

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| … | … | … |

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Legend:

Initially, the preference adjustment may seem costly and the number of evaluations may seem as large, as it is in the traditional PCs Matrix. However, this is not true. The solution to the problem with ten criteria and ten alternatives using RTM would require 145 evaluations (Table

The number of comparison indices is always greater than zero. The comparison indices of the main diagonal may not be evaluated, as each receives a value of 1. The comparison indices of the first row and first column of the comparisons matrix must be completed directly by the DM when he or she makes the PCs, where the first column is a direct result of this comparison.

As the PCs made by the DM are based on an interpretation of Saaty’s fundamental scale, the preference adjustment will also rely on this scale. A standard set of questions based on the fundamental scale of Saaty is proposed in Table

Proposed standard questions to assess consistency.

Indices Range | Question |
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| Considering the |

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| There was indecision between the previous question and the next question. |

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| Considering |

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| There was indecision between the previous question and the next question. |

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| Considering the |

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| There was indecision between the previous question and the next question. |

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| Considering the |

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| There was indecision between the previous question and the next question. |

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| Considering the |

As previously mentioned, to ensure an intuitive judgment by the DM, it is important to rank the alternatives from best to worst based on the criterion in question. Thus, the DM should only be asked to compare ordered alternatives in such a way that the best alternative is always positioned in the first row.

The assessment procedure by the DM is formalized considering the whole procedure of preference adjustment. A standard algorithm that structures all steps of the analysis was created (Figure

The first comparison index to be evaluated is always the one that compares the alternatives A2 and A3; it is

The second comparison index to be evaluated is always the one that compares alternatives A4 and A5.

Repeat this procedure until the nth alternative. If n is positive, the last comparison index necessarily represents the comparison between An-1 and An; also, An-1 does not change, since it has been previously evaluated (Table

If the answer is yes, go back to step

If the answer is no, proceed to step

If there was no change in terms of range, consider that the DM was given the opportunity to reassess the comparison index of the first line and that, despite his or her negative response in step

If there was a change in terms of range, go back to step

Preference adjustment algorithm.

An important issue to be considered is the automatic change in another comparison index when the DM revises and changes an index of the first row. Similarly, it is wise for researchers to consider that some indices of comparison, as assessed by the DM, will be subjected to change. When these changes occur, they must be reevaluated. However, the algorithm solves this problem.

If an index in the first row is changed, the other indexes that depend on it are automatically corrected considering the mathematical assumptions of the RTM. Thus, it will correct the deviations in the DM’s preferences and simultaneously maintain the consistency of the matrix.

As we did not perform evaluations with A1, start with

Without repeating the lines already evaluated, the DM must evaluate

Change in the comparisons matrix due to the preference adjustment of

A1 | A2 | A3 | A4 | A5 | A6 | |
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A1 | | | | | | |

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A2 | | | | | | |

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A3 | | | | | | |

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A4 | | | | | | |

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A5 | | | | | | |

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A6 | | | | | | |

Finally, no index containing A_{6} was evaluated and there is no alternative to pair with A_{6}. In this case, we must evaluate

Change in the comparisons matrix due to the preference adjustment of

A1 | A2 | A3 | A4 | A5 | A6 | |
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A1 | | | | | | |

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A2 | | | | | | |

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A3 | | | | | | |

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A4 | | | | | | |

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A5 | | | | | | |

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A6 | | | | | | |

We have applied the procedure for two situations. First, we used the problems explored in the experiment conducted on item 3 (tourist cities). Then, we tested the procedure with 10 DMs to determine the inconsistency and agreement level with the selected alternative.

We applied the proposed procedure to the case of selecting the next city to spend the summer holiday. The alternatives were Rio de Janeiro, Florianopolis, Salvador, and Natal, as presented in Figure

The procedure involved one DM and the support of an analyst to run the interactive algorithm. We will assume three criteria with the following weights: W1 = 0.60, W2 = 0.10, and W3 = 0.30, as previously defined.

Details of the procedure are presented with the Tourist Attractions criterion (Tables

Preference evaluation of the holiday destination choice problem for the criterion Tourist Attractions.

A1 | A2 | A3 | A4 | A5 | |
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A1 | 1.00 | 3.00 | 3.00 | 5.00 | 3.00 |

Additional preference information for the comparisons matrix of the holiday destination choice problem for the criterion Tourist Attractions.

A1 | A2 | A3 | A4 | A5 | |
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A1 | 1.00 | 3.00 | 3.00 | 5.00 | 3.00 |

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A2 | 0.33 | | 1.00 | 1.67 | 1.00 |

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A3 | 0.33 | 1.00 | 1.00 | 1.67 | 1.00 |

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A4 | 0.20 | 0.60 | 0.60 | 1.00 | 0.60 |

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A5 | 0.33 | 1.00 | 1.00 | | 1.00 |

Bold cell, representing the additional value required by DM.

Evaluate

Evaluate

Priority vectors of the alternatives for choosing the holiday destination.

Costs | Hotel Features | Tourist Attractions | ||||
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Weight | 0.60 | 0.10 | 0.30 | |||

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With adjustment | Without adjustment | Without adjustment | With adjustment | Without adjustment | With adjustment | |

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Rio de Janeiro | 0.071 | 0.053 | 0.506 | 0.466 | 0.455 | 0.484 |

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Florianopolis | 0.100 | 0.158 | 0.169 | 0.233 | 0.152 | 0.161 |

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Fortaleza | 0.166 | 0.158 | 0169 | 0.155 | 0.152 | 0.161 |

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Salvador | 0.166 | 0.158 | 0.101 | 0.093 | 0.091 | 0.097 |

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Natal | 0.489 | 0.474 | 0.056 | 0.052 | 0.152 | 0.097 |

The results presented in Table

Comparison of the problem’s results of choosing the holiday destination without preference adjustment and with preference adjustment.

Alternatives | Without adjustment | With adjustment | ||
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Priority Vector | Ranking | Priority Vector | Ranking | |

Rio de Janeiro | 0.243 | 2° | 0.223 | 2° |

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Florianopolis | 0.124 | 5° | 0.166 | 3° |

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Fortaleza | 0.162 | 3° | 0.159 | 4° |

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Salvador | 0.135 | 4° | 0.133 | 5° |

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Natal | 0.336 | 1° | 0.318 | 1° |

In the experiment, ten DMs of the very inconsistent group were chosen to test the PCs with preference adjustment. The DMs presented CR higher than 0.2, and thus their results were not considered in the experiment.

We begin the experiment reassessing the DM’s preferences for the first line of the matrix comparison. Then we apply the preference adjustment algorithm.

At the end of the process we asked two questions: Do you consider the effort and time spent applying the method to be reasonable? Do you consider that the result represents your preferences? For both questions, a five-point scale was used as an option for response:

For question 1, seven DMs responded, “Yes, for sure,” and three responded, “Usually, yes.” This indicates that effort declined considerably compared to the traditional method.

For question 2, five DMs responded, “Yes, for sure,” three responded, “Usually yes,” and two responded, “More or less.” This indicates that the results can be considered satisfactory in 80% of the cases.

The application of the proposed approach reveals that the effort required by the DMs to evaluate alternatives and to assess their preferences clearly decreased while attempting to maintain an acceptable level of inconsistency.

In this study, we demonstrate that the inconsistency of PC matrices, even within acceptable limits, can influence the results of a decision process. Thus, we proposed the following approach: First, the PC comprises only the first row of the matrix, while the other lines are filled in based on the assumptions of the RTM. Second, we use an algorithm to identify and correct deviations in the preferences of the DMs in the matrices.

The process of building a RTM, when applied to the PCs matrix, reduces the DM’s effort in the PCs. The significant reduction in a DM’s effort stems from the building of a RTM, which entails a totally consistent evaluation. Traditionally, PCs allow for certain level of inconsistency: CR less than or equal to 0.10.

The most important issue concerns the measurement of deviations when inconsistencies are allowed. The simulation and the experiments in this work clearly indicate that allowing inconsistencies achieves different results than when using consistent DM evaluations, although this is not always the case. This result does not nullify the value of building the RTM, nor does it discourage the use of a traditional PCs matrix.

Ultimately, the most important aspect is to check the consistency of the evaluation results to confirm that the DM’s priorities are reflected in his or her judgment. If the answer is negative, it may be necessary to reassess. In fact, this process is already included in virtually all MCDA methods, yet in the case of the PCs matrix, for which there are many alternatives, this would make decision making even more difficult. High cognitive efforts generate inconsistencies, in addition to requiring a significant amount of time. Thus, it is important to use tools that reduce cognitive effort and ensure satisfactory results.

The preference adjustment algorithm aims to complement the information provided by the DM. The adjustments approximate the results of the problem to the preferences of the DM. The procedures presented in this paper reduce the cognitive effort of the DM, eliminate inconsistencies in the comparison process, and present a recommendation that reflects the preferences of the DM.

More research is needed, however, to verify whether the results that are overly sensitive to inconsistency are linked to the scale used in the assessment, are caused by small differences between alternatives, or are simply the result of errors caused by the DM’s high cognitive effort. With regard to the application of the proposed algorithm, an important issue is to examine the use of other scales and the consistency index associated with these scales, such as the scale proposed by Koczkodaj [

The data included in the study "Exploring Multicriteria Elicitation Model Based on Pairwise Comparison: Building an Interactive Preference Adjustment Algorithm" are the responsibility of authors Giancarllo Ribeiro Vasconcelos and Caroline Maria de Miranda Mota, who collected the data in an open database. There are no restrictions on access to them.

The authors declare that they have no conflicts of interest.