Cognitive Best Worst Method for Multiattribute Decision-Making

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 thematrix based comparison style needs toomany comparisons and is not able to guarantee the consistency of thematrix when the number of objects involved is large. This research proposes a new MADMmethod 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 2n − 3 pairwise comparisons and is more likely to generate consistent comparisons and reliable results. The theoretical analysis and a real world application demonstrate the effectiveness of CBWM.

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 [19] and derives priority vector from the comparison matrix  = {  } whose entry   indicates the relative importance of object  dominating object .In AHP, the paired difference is measured by the ratio scale and correspondingly  is a pairwise reciprocal matrix (PRM).
Yuen [7,[20][21][22] pointed out that the paired ratio scale inappropriately represents the human perception or cognition of paired difference and potentially producing misapplications.Taking the comparison of two persons height, for example, if A is 1.4 m and B is 1.5 m, then we may say that B is slightly taller than A. However, if ratio scale is applied, the comparison result will be 2; for example, B is two times taller than A which is obviously unreasonable.The exaggeration of difference may cause the rank reversal problem and mislead the decision-maker.To handle this problem, Yuen proposed replacing the ratio scale by interval or differential scale which measures the difference rather than the quotient of the importance of two objects and accordingly converts the pairwise reciprocal matrix (PRM) to the pairwise opposite matrix (POM) and then derives the priority vector from POM by Primitive Cognitive Network Process (P-CNP).Apart from the scale problem, Rezaei [23,24] pointed out that the unstructured way in which pairwise comparisons are executed would cause the inconsistency of the pairwise comparison matrix.In order to improve the consistency, Rezaei proposed the Best Worst Method (BWM) which only compares each criterion to the best (most important, most desirable) and the worst (least important, least desirable) criterion, respectively.The pairwise comparison results are two vectors based on which a nonlinear model and a linear model are designed to derive the priority vector.BWM not only leads to more consistent comparisons but also reduces the number of comparisons.
Both P-CNP and BWM have been proved to be effective and been applied in some real world applications [22,[25][26][27][28][29][30][31][32][33][34].However, to the best of our knowledge, there has not been a pairwise comparison based MADM method that has the two aspects of advantages possessed by P-CNP and BWM at the same time.In this paper, we propose the Cognitive Best Worst Method (CBWM) which executes the pairwise comparisons in the same way as BWM and adopts the interval scale to measure the difference.CBWM possesses the advantages of both P-CNP and BWM and shows better characteristics than both the two methods.Compared to P-CNP, CBWM needs less pairwise comparisons and is likely to get more consistent pairwise comparisons.Compared to BWM, CBWM adopts the interval scale which is more suitable for measuring the cognitive paired comparisons and acquires more reliable results.The theoretical and experimental analyses demonstrate the good performance of CBWM.
The rest of this paper is organized as follows.Section 2 reviews P-CNP and BWM.Section 3 illustrates the Cognitive Best Worst Method.Section 4 presents an application using CBWM and compares CBWM to AHP, P-CNP, and BWM.Section 5 concludes the research.

Review of P-CNP and BWM
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 [7] and [23,24], respectively.

PCP.
In PCP, the goal of the decision-maker and the criteria set  = { 1 ,  2 , . . .,   } and the alternatives set  = { 1 ,  2 , . . .,   } are determined.Besides the elements aforementioned, a measurement scale schema (N, ) which maps the verbal scales N to the corresponding numerical scales  is also needed.As P-CNP adopts interval scales to measure the pairwise differences, the numerical representations should be in the following form: is the normal utility which is the mean of individual utility values of the comparison objects.2 + 1 is the number of intervals in the scale schema.From the definition of  we can get max() = .For the convenience of computation, we set  = ; then each element in  will be an integer.Table 2 gives an example of the ratio scale and the interval scale ( =  = 8).

CAP.
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  = (V 1 , V 2 , . . ., V  ); then the ideal POM will be B = { b }, where b = V  − V  .Let  = [  ] be the subjective judgment POM provided by the experts; then B is determined by  in the following way: The pairwise comparison matrix is called POM because   = −  holds for all ,  ∈ {1, 2, . . ., }.In [21], Yuen proposed that a POM  = [  ] is perfectly accordant if The accordance index (AI) is used to measure the degree of accordance.

CPP.
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 [22] pointed out that the closed form solution of model ( 5) is the Row Average plus the normal Utility (RAU) which can be expressed as If there exists V  < 0,  can be increased such that V  ≥ 0. LPS can be seen as the weighted version of PLS in which the weight is the penalty factor .The utility vector can be derived by the following model: After obtaining the utility vector  = {V 1 , V 2 , . . ., V  }, the normalized priority vector can be obtained easily by 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.
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.
Step 1. Determine a set of decision criteria.
Step 2. Determine the best criterion   and worst criterion   .
As is mentioned in [23], the decision-maker identifies the best and the worst criteria in general in this step.And if more than one criterion is considered to be the best or the worst, one can be chosen arbitrarily.No comparison is made at this stage.
Step 3. Execute pairwise comparisons between the best criterion and the other criteria.The result of the comparisons is the best-to-others vector denoted as Step 4. Execute pairwise comparisons between the other criteria and the worst criterion.The result of the comparisons is the others-to-worst vector denoted as Step 5. Derive the weight vector ( In (11),   is the weight of the best criterion;   is the weight of the worst criterion.Rezaei [24] pointed out that the solution of problem ( 11) is not unique when there are more than three criteria and the comparison vectors are not completely consistent and he proposed another linear model to derive the weight vector.min , Let  * be the solution of problem (12); the consistency ratio of the pairwise comparisons in BWM is defined as Consistency index in (13) is the maximal possible value of .Table 3 shows the consistency index for different values of   which is the preference of the best criterion over the worst criterion.

Cognitive Best Worst Method
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   = ( 1 ,  2 , . . .,   ) and   = ( 1 ,  2 , . . .,   ) to denote the best-to-others and others-to-worst vectors, while each element in the two vectors is expressed by interval scale as in Table 2.The best and the worst criteria should be determined by the decision-maker in the same way as mentioned in BWM.Let   be the best criterion and   be the worst criterion; then   denotes the difference between   and   , and   denotes the difference between   and   .
It is notable that all   and   are nonnegative.As   is the difference between the best criterion and the worst criterion, we can get that   is the maximal element in both   and   , and   ≤  ( is the normal utility and max() = ).Proposition 1.The pairwise comparison vectors   and   are perfectly accordant if or   +   =   , ∀ ∈ {1, 2, . . ., } .
Proof.Deduce ( 15) from (14).From (14) It is easy to verify that CI ≥ 0 and CI = 0 if and only if   and   are perfectly accordant.Now we discuss the relation of CI in (16) and AI in (4).The basic ideas of CI and AI are the same, for they both measure the average deviation of each element in the pairwise comparison matrix (vector) from consistency.The values of CI and AI are comparable for the same MADM problem.We define the random consistency index CI  as the mathematical expectation of CI when all the values except   in   and   are selected randomly and independently from the set {0, 1, . . .,   }.

Theorem 3. Consider the following:
Proof.Let Δ  = ((  +  −  )/) 2 .When the th criterion is the best or the worst one, the value of Δ  will be zero; otherwise the value of Δ  will be in the closed interval [0, (  /) 2 ] and the corresponding probability distribution is given in Table 4.It is easy to verify that ∑ According to the probability distribution of Δ  , we can get the mathematical expectation of Δ  .

𝐸 (Δ
The values corresponding to different criteria in the two vectors   and   are independent, so we can get For AI, we define random accordance index AI  as the mathematical expectation of AI when all the elements of the pairwise opposite matrix are selected from the set {−, . . ., −1, 0, 1, . . ., } randomly and independently under the constraint that   = −  .To get the value of AI  , we run the simulation which generates the POM in the way as just described 10000 times and calculate AI each time; then we take the average of the 10000 values as the estimate of AI  . Table 5 shows the values of CI  and AI  with different  and  when   = .From Table 5 we can clearly see that AI  > CI  for the given  and  which means that CBWM is more likely to get a more consistent pairwise comparison result than P-CNP.The reason of this phenomenon is that the elements in vectors of CBWM are all nonnegative while the elements of the POM may be either positive or negative, which leads to the fact that the maximal value of ((  +   −   )/) 2 is 1 and the maximal value of ((  +   −   )/) 2 is 9.
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 decisionmaker 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 6.The first and the second rows are the values of   and   , respectively.We can sum the elements in rows 1 and 2 and compare the results to   .If   +   is too different from   , we should revise   and   to get more consistent vectors.

Derivation of Utility Vector
Theorem 4. When the comparison vectors   and   are completely consistent, the utility vector  = (V 1 , V 2 , . . ., V  ) can be derived by or Proof.When   and   are completely consistent, we can get When the comparison vectors   and   are not fully consistent, we propose two methods to derive the utility vector: the minimization of the maximal deviation method (MMD) and the minimization of total deviation method (MTD).
In MMD, the optimal utility vector  = (V 1 , V 2 , . . ., V  ) is the one where, for each pair of V  −V  and V  −V  , we have V  − V  =   and V  −V  =   .To satisfy these conditions, we find a solution where the maximal absolute differences |V  −V  −  | and |V  − V  −   | for all  ∈ {1, 2, . . ., } are minimized.The optimal utility vector can be obtained by solving the following problem: Problem ( 22) can be transferred to the following linear programming problem: min , In MTD, the optimal utility vector should be the one where the total deviation Analogous to [22], we can get the closed form solution of model ( 24) as follows: Also, if there exists V  < 0,  can be increased such that V  ≥ 0.

Example 2.
Here we revise Example 1 in [23] using CBWM to illustrate the CBWM steps discussed above.The example involves selecting a transport mode for a company.The criteria determined by the company are as follows:  1 : load flexibility,  2 : accessibility, and  3 : cost.The company identifies that  3 is the best criterion and  1 is the worst criterion, and the original comparison vectors in ratio scale are   = (8, 2, 1) and   = (1, 5, 8).In interval scale, the corresponding comparison vectors will be   = (7, 1, 0) and   = (0, 4, 7) with  = 8.Firstly, we calculate the consistency index by ( 16) and we can get CI = 0.144.To get the weight vector, we substitute the data into model (23) and model (24) and solve the two problems.

Illustrations
In this section, we compare the proposed MADM method CBWM to AHP, P-CNP, and BWM on a real world MADM problem.

Problem Description.
Here the transport company selection problem discussed by [7,35] is revised by AHP, P-CNP, BWM, and CBWM methods.A company needs to select a freight transportation company from four alternatives (T1, T2, T3, and T4) to deliver its goods.The criteria considered are transportation costs (TC), defective rate (DR), tardiness rate (TR), flexibility (F), and documentation ability (DA).
Six pairwise reciprocal matrices are shown in Table 7.The top left matrix is the comparisons between criteria and the other five matrices are comparisons between alternatives with respect to different criteria.Saaty's consistency ratio is given under each matrix.
Table 8 shows six pairwise opposite matrices ( = 8) under interval scale corresponding to the six matrices in Table 7.The accordance index (AI) defined in ( 4) is given under each matrix.
Tables 9 and 10 show the pairwise comparison vectors under ratio and interval scale, respectively.These vectors are extracted from the corresponding matrices.Best # represents the index of the best criterion (alternative) and Worst # represents the index of the worst criterion (alternative).It is worth noting that, on criterion TC, the alternatives T1 and T3 are equal and are both the worst.Rezaei [23] pointed out that if more than one criterion is considered to be the best or the worst, one can be chosen arbitrarily.Here we choose T1 as the worst with respect to TC.
The comparison matrices on criteria DR and DA are completely consistent in Table 8 while it is not the case in Table 7.The phenomenon indicates the consistent cognition for pairwise comparison in a matrix can be distorted as inconsistent by ratio scale [7].It is also the same when replacing the comparison matrices with comparison vectors.We can see that the comparison vectors on criteria DR and DA are completely consistent in Table 10 while it is not the case in Table 9.It is also worth noting that the pairwise comparison vectors are always consistent when the corresponding matrices are consistent under whatever scales while the opposite is obviously not the case.As per the results above and the analysis of consistency index in Section 3.1, we can say that CBWM can get more consistent pairwise comparisons than AHP, BWM, and P-CNP.show the prioritization and aggregation results along with the ranking of the alternatives from the best to the worst.The results of AHP in Table 11 are obtained by the eigenvector methods.The results of BWM in Table 12 are obtained by solving linear model (12).The results of P-CNP in Table 13 are obtained by the PLS method in (5).The results of CBWM-MMD in Table 14 are obtained by the MMD method in (23).The results of CBWM-MTD in Table 15 are obtained by the MTD method in (24).

Results of Different Methods.
The results in [7] show that P-CNP performs better than AHP because P-CNP gets stable rankings with different prioritization methods while AHP get fluctuant rankings with different prioritization methods.From our results in Tables 11-15, we can see that the BWM obtains similar results to AHP with respect to both ranking and total utility, while the two prioritization methods of CBWM both achieve results very close to the P-CNP's results.We can say that CBWM acquires more reliable results and performs better than BWM in this case based on the conclusion in [7].

4.3.
Comparisons with respect to Total Deviation.In order to measure the performance of different methods further, we employ total deviation (TD) [36] to make the comparison.The original definition of total deviation (TD) is the Euclidean distance between the ratios of weights   /  and their pairwise comparison PRM(, ).
PRM is the pairwise reciprocal matrix using ratio scale;   and   are the weights of the th criterion and the th criterion, respectively.For interval scale, we need to modify the definition of TD as POM is the pairwise opposite matrix using interval scale; V  and V  are the utilities of the th criterion and the th criterion, respectively.
As the numbers of pairwise comparisons of P-CNP (AHP) and CBWM (CWM) are  2 and 2, respectively, it is    ) , Table 16 shows the TDs of different methods when determining the relative importance of each criterion.From the table, we can see that the total deviations of ratio scale based methods are significantly bigger than interval scale based methods; that is to say, the proposed CBWM performs better than BWM.It is worth noting that the TD of P-CNP is smaller than CBWM, but it does not mean that P-CNP performs better than CBWM.The reason is that the accordance index of P-CNP is 0.1168 while the consistency index of CBWM is 0.2016.According to the analysis of the relationship of CI and AI in Section 3.1, the phenomenon that AI < CI is unusual.

Conclusion and Future Research
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: (1) CBWM is vector based method and needs less pairwise comparisons than matrix based methods.The numbers of pairwise comparisons are 2 − 3 and ( − 1)/2 for CBWM and matrix based methods, respectively, where  is the number of objects involved in the comparisons.
(2) CBWM can get more consistent pairwise comparison results due to the use of interval scale and the best worst comparison strategy.
(3) 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.

Table 2 :
Scale schemas: ratio scales and interval scales.

Table 4 :
The probability distribution of Δ  .

Table 5 :
CI  and AI  with different  and .
we can get the weight vector  Supposing, in a MADM problem, there are four criteria { 1 ,  2 ,  3 ,  4 }, among which  1 is the best criterion and  2 is the worst criterion.The comparison vectors provided by the experts are  to get the utility vector.The utility vector derived from (20) is  =(12.74,4.75,7.75, 6.75) and the corresponding weight vector is  = (0.398, 0.148, 0.242, 0.212).

Table 7 :
Pairwise comparison matrices under ratio scale.

Table 8 :
Pairwise comparison matrices under interval scale.

Table 9 :
Pairwise comparison vectors under ratio scale.

Table 10 :
Pairwise comparison vectors under interval scale.

Table 11 :
Results of AHP.

Table 16 :
TDs of different methods.to judge the performances of these methods directly by TD.To make fair comparison, we employ the average TD which is the quotient of MV divided by the number of pairwise comparisons.So the definitions of TD for each method would be unfair