Pairwise comparison based multiattribute decision-making (MADM) methods are widely used and studied in recent years. However, the perception and cognition towards the semantic representation for the linguistic rating scale and the way in which the pairwise comparisons are executed are still open to discuss. The commonly used ratio scale is likely to produce misapplications and the matrix based comparison style needs too many comparisons and is not able to guarantee the consistency of the matrix when the number of objects involved is large. This research proposes a new MADM method CBWM (Cognitive Best Worst Method) which adopts interval scale to represent the pairwise difference and only compares each object to the best object and the worst object rather than all the other objects. CBWM is a vector based method which only needs
A typical multiattribute decision-making (MADM) problem can be described in Table
Description of MADM problem.
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Among these methods, AHP has been increasingly studied by plenty of researchers and applied in various applications due to its simplicity and practicability. AHP is based on pairwise comparison which is first introduced by Thurstone [
Yuen [
Apart from the scale problem, Rezaei [
Both P-CNP and BWM have been proved to be effective and been applied in some real world applications [
The rest of this paper is organized as follows. Section
In this section, we briefly review the concept and process of P-CNP and BWM. For the detailed description of the two methods, we refer to [
The P-CNP is the cognitive architecture which comprises cognitive decision processes: Problem Cognitive Process (PCP), Cognitive Assessment Process (CAP), Cognitive Prioritization Process (CPP), Multiple Information Fusion Process (MIP), and Decision Volition Process (DVP).
In PCP, the goal of the decision-maker and the criteria set
Scale schemas: ratio scales and interval scales.
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Verbal scales | Ratio scales | Interval scales |
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0 | Equally | 1 | 0 |
1 | Weakly | 2 | 1 |
2 | Moderately | 3 | 2 |
3 | Moderately plus | 4 | 3 |
4 | Strongly | 5 | 4 |
5 | Strongly plus | 6 | 5 |
6 | Very strongly | 7 | 6 |
7 | Very, very strongly | 8 | 7 |
8 | Extremely | 9 | 8 |
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Reciprocals/opposites of the above | (from 1/9 to 1) | (from −8 to 0) |
In CAP, the pairwise comparisons are executed and the pairwise opposite matrix (POM) is constructed. Let an ideal utility vector of the objects involved in the pairwise comparisons be
The pairwise comparison matrix is called POM because
The accordance index (AI) is used to measure the degree of accordance.
If
The purpose of CPP is to derive the priority vector from the POM. The two common used methods are primitive least squares (PLS) method and least penalty squares (LPS) method.
The PLS method derives the utility vector through the following model:
Yuen [
If there exists
LPS can be seen as the weighted version of PLS in which the weight is the penalty factor
After obtaining the utility vector
MIP is the same as the aggregation process traditionally mentioned in MADM and DVP is the decision process based on the aggregated global utility of each alternative.
BWM is a vector based MADM method which requires fewer comparisons than the matrix based method such as AHP. Here we briefly describe the steps of BWM.
Determine a set of decision criteria.
Determine the best criterion
As is mentioned in [
Execute pairwise comparisons between the best criterion and the other criteria. The result of the comparisons is the best-to-others vector denoted as
Execute pairwise comparisons between the other criteria and the worst criterion. The result of the comparisons is the others-to-worst vector denoted as
Derive the weight vector
In (
Let
Consistency index in (
Consistency index (CI) table of BWM.
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1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
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CI | 0 | 0.44 | 1.00 | 1.63 | 2.30 | 3.00 | 3.73 | 4.47 | 5.23 |
In this section, we elaborate the proposed Cognitive Best Worst Method (CBWM) which can be seen as interval ratio based BWM. The basic process of CBWM is the same as BWM, so we focus on the differences caused by introducing interval ratio to represent pairwise difference.
We still use
The pairwise comparison vectors
Two conditions (
Deduce (
Based on Proposition
It is easy to verify that
Now we discuss the relation of
Consider the following:
Let
According to the probability distribution of
The values corresponding to different criteria in the two vectors
The probability distribution of
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For
Table
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3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |
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0.279 | 0.342 | 0.374 | 0.394 | 0.408 | 0.418 | 0.426 | 0.432 | 0.437 |
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0.341 | 0.520 | 0.629 | 0.703 | 0.756 | 0.797 | 0.827 | 0.852 | 0.873 | |
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0.272 | 0.333 | 0.365 | 0.385 | 0.398 | 0.408 | 0.416 | 0.422 | 0.426 |
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0.337 | 0.512 | 0.621 | 0.694 | 0.746 | 0.785 | 0.816 | 0.841 | 0.861 | |
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0.267 | 0.327 | 0.359 | 0.378 | 0.391 | 0.401 | 0.408 | 0.414 | 0.419 |
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0.335 | 0.508 | 0.615 | 0.686 | 0.738 | 0.777 | 0.807 | 0.831 | 0.852 | |
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0.264 | 0.323 | 0.354 | 0.373 | 0.386 | 0.295 | 0.403 | 0.408 | 0.412 |
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0.330 | 0.504 | 0.609 | 0.681 | 0.731 | 0.770 | 0.801 | 0.825 | 0.846 | |
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0.261 | 0.319 | 0.350 | 0.369 | 0.381 | 0.391 | 0.398 | 0.404 | 0.408 |
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0.330 | 0.501 | 0.605 | 0.677 | 0.727 | 0.766 | 0.795 | 0.820 | 0.840 |
The above analysis is valid under the circumstance that the decision-maker is able to appropriately determine the best and the worst criteria. Comparing the best and the worst objects to the others matches the habit and process of human cognition and then is likely to get more consistent results than comparing each pair of the objects. However, if the decision-maker is not able to determine the best and the worst criteria, the above analysis is of no sense and CBWM method is not suitable for the problem.
Another advantage of CBWM is that it is easy to check the consistency and identify the cause of inconsistency. The examination and identification can be done intuitively with the help of Table
Consistency check (CC) table.
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CC |
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When the comparison vectors
When
When the comparison vectors
In MMD, the optimal utility vector
Problem (
In MTD, the optimal utility vector should be the one where the total deviation
Analogous to [
Also, if there exists
Solving problem (
Supposing, in a MADM problem, there are four criteria
Here we revise Example 1 in [
Firstly, we calculate the consistency index by (
By MMD, we can get the utility vector
By MTD, we can get the utility vector
In this section, we compare the proposed MADM method CBWM to AHP, P-CNP, and BWM on a real world MADM problem.
Here the transport company selection problem discussed by [
Six pairwise reciprocal matrices are shown in Table
Pairwise comparison matrices under ratio scale.
Criteria | TC | DR | TR | F | DA | TC | T1 | T2 | T3 | T4 |
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TC | 1 | 5 | 3 | 5 | 9 | T1 | 1 | 1/3 | 1 | 1/5 |
DR | 1/5 | 1 | 1/2 | 1/2 | 7 | T2 | 3 | 1 | 3 | 1/2 |
TR | 1/3 | 2 | 1 | 1/2 | 7 | T3 | 1 | 1/3 | 1 | 1/5 |
F | 1/5 | 2 | 2 | 1 | 8 | T4 | 5 | 2 | 5 | 1 |
DA | 1/9 | 1/7 | 1/7 | 1/8 | 1 | |||||
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DR | T1 | T2 | T3 | T4 | TR | T1 | T2 | T3 | T4 | |
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T1 | 1 | 7 | 3 | 5 | T1 | 1 | 1/5 | 1/5 | 2 | |
T2 | 1/7 | 1 | 1/5 | 1/3 | T2 | 5 | 1 | 1/3 | 7 | |
T3 | 1/3 | 5 | 1 | 3 | T3 | 5 | 3 | 1 | 7 | |
T4 | 1/5 | 3 | 1/3 | 1 | T4 | 1/2 | 1/7 | 1/7 | 1 | |
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F | T1 | T2 | T3 | T4 | DA | T1 | T2 | T3 | T4 | |
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T1 | 1 | 5 | 1/3 | 3 | T1 | 1 | 1/5 | 1/3 | 1/3 | |
T2 | 1/5 | 1 | 1/7 | 1/3 | T2 | 5 | 1 | 3 | 3 | |
T3 | 3 | 7 | 1 | 7 | T3 | 3 | 1/3 | 1 | 1 | |
T4 | 1/3 | 3 | 1/7 | 1 | T4 | 3 | 1/3 | 1 | 1 | |
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Table
Pairwise comparison matrices under interval scale.
Criteria | TC | DR | TR | F | DA | TC | T1 | T2 | T3 | T4 |
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TC | 0 | 4 | 2 | 4 | 8 | T1 | 0 | −2 | 0 | −4 |
DR | −4 | 0 | −1 | −1 | 6 | T2 | 2 | 0 | 2 | −1 |
TR | −2 | 1 | 0 | −1 | 6 | T3 | 0 | −2 | 0 | −4 |
F | −4 | 1 | 1 | 0 | 7 | T4 | 4 | 1 | 4 | 0 |
DA | −8 | −6 | −6 | −7 | 0 | |||||
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DR | T1 | T2 | T3 | T4 | TR | T1 | T2 | T3 | T4 | |
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T1 | 0 | 6 | 2 | 4 | T1 | 0 | −4 | −4 | 1 | |
T2 | −6 | 0 | −4 | −2 | T2 | 4 | 0 | −2 | 6 | |
T3 | −2 | 4 | 0 | 2 | T3 | 4 | 2 | 0 | 6 | |
T4 | −4 | 2 | −2 | 0 | T4 | −1 | −6 | −6 | 0 | |
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F | T1 | T2 | T3 | T4 | DA | T1 | T2 | T3 | T4 | |
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T1 | 0 | 4 | −2 | 2 | T1 | 0 | −4 | −2 | −2 | |
T2 | −4 | 0 | −6 | −2 | T2 | 4 | 0 | 2 | 2 | |
T3 | 2 | 6 | 0 | 6 | T3 | 2 | −2 | 0 | 0 | |
T4 | −2 | 2 | −6 | 0 | T4 | 2 | −2 | 0 | 0 | |
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Tables
Pairwise comparison vectors under ratio scale.
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Criteria | 1 | (1, 5, 3, 5, 9) | 5 | (9, 7, 7, 8, 1) |
TC | 4 | (5, 2, 5, 1) | 1 | (1, 3, 1, 5) |
DR | 1 | (1, 7, 3, 5) | 2 | (7, 1, 5, 3) |
TR | 3 | (5, 3, 1, 7) | 4 | (2, 7, 7, 1) |
F | 3 | (3, 7, 1, 7) | 2 | (5, 1, 7, 3) |
DA | 2 | (5, 1, 3, 3) | 1 | (1, 5, 3, 3) |
Pairwise comparison vectors under interval scale.
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Criteria | 1 | (0, 4, 2, 4, 8) | 5 | (8, 6, 6, 7, 0) |
TC | 4 | (4, 1, 4, 0) | 1 | (0, 2, 0, 4) |
DR | 1 | (0, 6, 2, 4) | 2 | (6, 0, 4, 2) |
TR | 3 | (4, 2, 0, 6) | 4 | (1, 6, 6, 0) |
F | 3 | (2, 6, 0, 6) | 2 | (4, 0, 6, 2) |
DA | 2 | (4, 0, 2, 2) | 1 | (0, 4, 2, 2) |
The comparison matrices on criteria DR and DA are completely consistent in Table
Tables
Results of AHP.
AHP | TC | DR | TR | F | DA | Result | Rank |
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W | 0.513 | 0.108 | 0.156 | 0.195 | 0.027 | ||
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T1 | 0.099 | 0.565 | 0.087 | 0.249 | 0.078 | 0.176 | 4 |
T2 | 0.284 | 0.055 | 0.312 | 0.054 | 0.522 | 0.225 | 3 |
T3 | 0.099 | 0.262 | 0.549 | 0.592 | 0.200 | 0.286 | 2 |
T4 | 0.518 | 0.118 | 0.053 | 0.105 | 0.200 | 0.313 | 1 |
Results of BWM.
BWM | TC | DR | TR | F | DA | Result | Rank |
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W | 0.488 | 0.130 | 0.216 | 0.130 | 0.036 | ||
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T1 | 0.099 | 0.572 | 0.144 | 0.235 | 0.089 | 0.187 | 4 |
T2 | 0.272 | 0.067 | 0.240 | 0.069 | 0.518 | 0.221 | 3 |
T3 | 0.109 | 0.226 | 0.559 | 0.595 | 0.196 | 0.288 | 2 |
T4 | 0.520 | 0.136 | 0.057 | 0.101 | 0.196 | 0.304 | 1 |
Results of P-CNP.
P-CNP | TC | DR | TR | F | DA | Result | Rank |
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W | 0.290 | 0.200 | 0.220 | 0.225 | 0.065 | ||
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T1 | 0.203 | 0.344 | 0.195 | 0.281 | 0.188 | 0.246 | 2 |
T2 | 0.273 | 0.156 | 0.313 | 0.156 | 0.313 | 0.235 | 3 |
T3 | 0.203 | 0.281 | 0.344 | 0.359 | 0.250 | 0.288 | 1 |
T4 | 0.320 | 0.219 | 0.148 | 0.203 | 0.250 | 0.231 | 4 |
Results of CBWM-MMD.
CBWM-MMD | TC | DR | TR | F | DA | Result | Rank |
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W | 0.292 | 0.201 | 0.224 | 0.217 | 0.067 | ||
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T1 | 0.204 | 0.344 | 0.199 | 0.284 | 0.188 | 0.247 | 2 |
T2 | 0.277 | 0.156 | 0.309 | 0.155 | 0.313 | 0.236 | 3 |
T3 | 0.199 | 0.281 | 0.348 | 0.364 | 0.250 | 0.288 | 1 |
T4 | 0.319 | 0.219 | 0.145 | 0.197 | 0.250 | 0.229 | 4 |
Results of CBWM-MTD.
CBWM-MTD | TC | DR | TR | F | DA | Result | Rank |
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W | 0.286 | 0.203 | 0.228 | 0.215 | 0.069 | ||
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T1 | 0.202 | 0.344 | 0.199 | 0.289 | 0.188 | 0.248 | 2 |
T2 | 0.277 | 0.156 | 0.309 | 0.159 | 0.313 | 0.237 | 3 |
T3 | 0.199 | 0.281 | 0.342 | 0.357 | 0.250 | 0.286 | 1 |
T4 | 0.322 | 0.219 | 0.150 | 0.195 | 0.250 | 0.230 | 4 |
The results in [
In order to measure the performance of different methods further, we employ total deviation (TD) [
For interval scale, we need to modify the definition of TD as
As the numbers of pairwise comparisons of P-CNP (AHP) and CBWM (CWM) are
Table
TDs of different methods.
Methods | AHP | BWM | P-CNP | CBWM-MMD | CBWM-MTD |
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TD | 4.534 | 7.532 | 0.416 | 0.513 | 0.471 |
In this paper, a new interval scale and vector based pairwise comparison MADM method CBWM (Cognitive Best Worst Method) is proposed. CBWM combines the ideas of P-CNP and BWM method and shows better characteristics than the two methods. The theoretical and experimental analyses demonstrate the good performance of CBWM. The excellent features CBWM shows are summarized as follows: CBWM is vector based method and needs less pairwise comparisons than matrix based methods. The numbers of pairwise comparisons are CBWM can get more consistent pairwise comparison results due to the use of interval scale and the best worst comparison strategy. As the interval scale is more suitable for measuring cognitive paired comparisons, CBWM acquires more reliable decision results than ratio scale based methods.
It is worth noting that CBWM is only suitable for the problem in which the decision-maker is able to determine the best and the worst criteria.
The future research will focus on two aspects; one is to apply CBWM to more real world MADM problems, and the other is to extent CBWM to fuzzy or interval judgments to capture the uncertainty in the process of MADM.
The authors declare that they have no conflicts of interest.
This work was supported by the Natural Science Foundation of Jiangsu Province (Grants no. BK20150720).