An Approach for Generating Weights Using the Pairwise Comparison Matrix

Data envelopment analysis (DEA) has been used for obtaining weights for the analytic hierarchy process (AHP), an approach known as DEAHP. +is method sometimes identifies more than one decision criterion or alternative as DEAHP-efficient. To overcome this problem, this paper proposes a new approach that not only generates appropriate weights for the decision criteria or alternatives, but also differentiates between DEAHP-efficient decision criteria or alternatives. To this end, we propose a DEA model with an assurance region and a cross-weight model that prioritizes decision criteria or alternatives by considering their most unfavorable weights. Two numerical examples are also provided to illustrate the advantages and potential applications of the proposed model.


Introduction
An important issue in the analytic hierarchy process (AHP) is obtaining a priority vector from a pairwise comparison matrix, which has widely been investigated in AHP articles [1][2][3]. erefore, the key issue is how to determine the weights of decision criteria or alternatives [4][5][6]. Since the advent of this technique, it has been theoretically developed multiple times and applied for various purposes (e.g., review studies of Kheybari et al. [7] and Emrouznejad and Marra [8]).
Several methods have been proposed in AHP-related articles for the determination of local weights based on the pairwise comparison matrices [9][10][11][12][13][14][15][16][17][18]. Each of these methods has specific advantages and disadvantages, and thus, none of them can be considered the best. Although the eigenvector method (EM) has been strongly recommended and preferred by Saaty [19], there is no general agreement on its superiority (e.g., Mikhailov [20] and Srdjevic [21]). e data envelopment analysis (DEA) method, proposed by Charnes, Cooper [22], has been used for prioritization in AHP. In this approach, decision criteria or alternatives in a pairwise comparison matrix are considered decision-making units (DMUs). Moreover, the elements on the rows of the pairwise comparison matrix are the outputs of DMUs, and the efficiency of DMUs is considered priorities of the pairwise comparison matrix. Accordingly, Ramanathan [23] proposed a DEAHP method to obtain both weights and cumulative weights in AHP. Although DEAHP can generate true weights for consistent pairwise comparison matrices, it generates illogical and meaningless weights when it comes to inconsistent pairwise comparison matrices. Wang, Chin [24] and Wang, Chin [25] detailed the disadvantages of DEAHP by providing numerical examples. To overcome these issues, they proposed a DEA model with an assurance region (AR) to generate weights in the AHP. e DEA/AR model can generate intuitive and even logical weights for both consistent and inconsistent pairwise comparison matrices. Wang, Parkan [26] proposed a linear programming method to generate the most desirable weights from pairwise comparison matrices. Wang and Chin [27] proposed a DEA model for prioritization in AHP. Instead of the efficiency of each DMU, the proposed model defines relative efficiency as its priority. As a result, it generates the most desirable weights, which are close to the weights of the eigenvector of the pairwise comparison matrices. e most desirable weights for each decision criterion or alternative are assessed from its view, which is called selfevaluation [28,29]. When a decision criterion or alternative evaluates its best weight, it also evaluates other decision criteria or alternatives. Such weights, evaluated using other criteria or alternatives, are called cross-weights [30]. e evaluations done by other criteria or alternatives are called peer-evaluation. Obviously, cross-weights may not be desirable for decision criteria or alternatives. erefore, the use of the most desirable weights is not suitable for a comprehensive decision-making.
To generate a comprehensive weight that is also logical for every decision criterion or alternative, Wang, Luo [28] proposed the cross-weight evaluation technique for weight differentiation.
e cross-weight evaluation assessed the weights of a pairwise comparison matrix not only from its standpoint, but also from a peer standpoint. erefore, the weights generated from the cross-weight evaluation technique are more logical and fair. e models proposed by Wang et al. [28] were among the optimistic models in DEA. Although those models are more optimal than available DEAHP models in differentiation, similar to other DEAHP models, they may sometimes evaluate some decision criteria or alternatives at the same level, making decision-making problematic. e present paper proposed a new approach, called "analysis of the most undesirable weight," to obtain the weights of decision criteria and alternatives. It enables us to rank a set of decision criteria and alternatives, even those assessed at the same level from an optimistic view. e principal condition for the analysis of the most undesirable weight is that, in addition to the most desirable weight, the most undesirable weight can also be assigned to every decision criterion and alternative. e most desirable and undesirable weights for a decision criterion or alternative define the weight interval of that criterion or alternative.
Two numerical examples will be provided to show the advantages and potential applications of the proposed model. e remaining parts of the paper are organized as follows: Section 2 briefly reviews the DEAHP models for the generation of the most desirable weights for pairwise comparison matrices. Section 3 deals with the cross-weight evaluation technique from a pessimistic standpoint. Section 4 presents numerical examples. Section 5 presents comparison with other prioritization methods. Section 6 is devoted to the conclusion.

DEAHP Models for Generating Most Desirable Weights
Assume that A � a ij n×n � a 11 a 12 · · · a 1n a 21 a 22 · · · a 2n ⋮ ⋮ · · · ⋮ a n1 a n2 · · · a nn is a pairwise comparison matrix with a ii � 1 and a ji � 1/a ij for j ≠ i and W � (w 1 , . . . , w n ) T is its priority vector. In DEAHP, each row of A is considered a DMU, and each column is considered an output. Accordingly, Wang, Chin [25] proposed DEA model (2) to generate weights from pairwise comparison matrices A: where o refers to the criterion or alternative under the evaluation. By solving model (2) for each w o (o � 1, . . . , n), the most desirable weights for n decision criteria or alternatives can be obtained. v j (j � 1, . . . , n) are decision variables, and w j /β ≤ v j ≤ w j /n (j � 1, . . . , n) are the AR imposed on the DEA model. β is the upper boundary of the maximum eigenvector of the pairwise comparison matrix A � (a ij ) n×n , which is determined from the following equation: where r 1 , . . . , r n and c 1 , . . . , c n are, respectively, the sums of rows and columns of A � (a ij ) n×n . If there is a set of positive weights v * j , (j � 1, . . . , n) to fulfill w * o � 1, the decision criterion or alternative is called DEAHP efficient; otherwise, it is called DEAHP nonefficient. Typically, the best decision criterion or alternative can be selected from DEAHP-efficient decision criteria or alternatives.
Another technique for generating weight in the DEAHP is the use of cross-weight evaluation to obtain priorities. Model (2) only considers the most desirable weights obtained from self-evaluation and may generate several optimal weight vectors. To solve the problem of nonuniqueness, Wang et al. [28] proposed the cross-weight model (4) that generates unique weights: where w * o is the most desirable weight of criterion or alternative under investigation based on model (2).

DEAHP Models for Generating Most Undesirable Weights
Model (2) may recognize more than one decision criterion or alternative as efficient. ese efficient decision criteria or alternatives cannot be differentiated based on their most desirable weights. However, we can consider the most undesirable weights. e most undesirable weight is in contrast to the most desirable weight and represents the weight of each decision criterion or alternative in the most undesirable condition. eoretically, the most desirable and undesirable weights should be obtained in the same range, which should present an interval for each decision criterion or alternative. For example, they can be measured in the interval of [α, 1], where α > 0 is a predefined parameter [31][32][33]. e corresponding model can be created as follows: Nevertheless, if the most undesirable weight is measured in this way, the major problem is the need to predefine an appropriate value for α. To avoid the problem of determining the value of α, the most undesirable weight of each decision criterion or alternative can be measured with the following model: is model can be solved for all considered decision criteria and alternatives. e decision criterion or alternative with the greatest weight can be selected. e advantage of model (6) relative to model (2) is that it allows for differentiating between decision criteria and alternatives of DEAHP efficient. If there was a set of positive weights for which the value of an optimal objective function of model (6) is 1 or w * o � 1, it would be said that the decision criterion or alternative is DEAHP inefficient; otherwise, it is called DEAHP non-inefficient. Theorem 1. If A � (a ij ) n×n was a perfectly consistent pairwise comparison matrix, then model (6) would generate the following weights: which are the normalization of true weights Proof. Since A � (a ij ) n×n is a perfectly consistent pairwise comparison matrix, it can be recognized by the weights of . . . , w n }. Model (6) only considers the most undesirable weights obtained from self-evaluation and may generate several optimal weight vectors. To solve the problem of nonuniqueness, cross-weight model (8), which generates unique weights, is recommended: where w * o is the most undesirable weight of criterion or alternative under investigation, based on model (6).

Numerical Examples
In this part, two numerical examples were investigated using the proposed approach to show its application and capability in the differentiation of efficient decision criteria or alternatives.

Journal of Mathematics
Example 1. Consider the following pairwise comparison matrix of Wang and Chin [27]: is pairwise comparison matrix has a good consistency CR � 0.0999. β � 6.3810 was obtained for this pairwise comparison matrix. Table 1 shows the priorities obtained from EM and DEA/AR models (2) and (6). According to Table 1, it can be seen that EM and the DEA/AR model (6) evaluate C 3 as the most important criterion or alternative. DEA/AR models (2) and (6) differ in the evaluation of C 3 and C 4 . e DEA/AR model (2) assesses C 3 and C 4 at the same level (defines C 3 and C 4 as efficient) and cannot differentiate between them, while the DEA/AR model (6) can easily differentiate C 3 from C 4 , and evaluates C 3 as more important than C 4 . In other words, DEA/AR model (6) provides a general rank of decision criteria or alternatives. Tables 2 and 3 show the results from the evaluation of cross-weights obtained from DEA/AR models (2), (4), (6), and (8). Obviously, the mean cross-weights generated using the DEA/AR models (2) and (4) (Table 2) indicate that C 3 and C 4 are at the same level of importance, whereas the mean cross-weights generated using the DEA/AR models (6) and (8) (Table 3) indicate that C 3 is more important than C 4 . erefore, the mean cross-weights obtained from DEA/AR models (6) and (8) have greater differentiation performance than the mean cross-weights obtained from DEA/AR models (2) and (3).

Example 2.
Consider the supplier selection problem of BEKO (a major TV manufacturer in Turkey), derived from Sevkli et al. [34]. e hierarchical structure of the supplier selection problem for purchasing CRTs is presented in Figure 1. It consists of six evaluation criteria, 25 subcriteria, and three suppliers. e six evaluation criteria are as follows: C 1 : performance assessment C 2 : human resources C 3 : quality system assessment C 4 : manufacturing C 5 : business criteria C 6 : using information technology Each criterion is divided into 3-6 subcriteria, forming a total of 25 subcriteria. ese 25 subcriteria are as follows: C 11 : shipment Quality C 12 : delivery C 13 : cost analysis C 21 : number of Employees C 22 : organizational structure C 23 : training C 24 : number of technical staff C 31 : management commitment C 32 : inspection C 33 : quality planning C 34 : quality assurance C 41 : production capacity C 42 : predictive and preventive maintenance C 43 : lead-time C 44 : up-to-date techniques and equipment C 45 : transportation-storage and packaging C 46 : new product development C 51 : reputation C 52 : geographic location C 53 : price C 54 : patent C 55 : technical capacity C 61 : radio-frequency identification (RFID) C 62 : electronic data interchange (EDI) C 63 : internet For anonymity, the three supplier companies are named by numbers 1, 2, and 3. In Tables 4-16, the pairwise comparison matrix has been provided for six evaluation criteria, 25 subcriteria, and three suppliers. Moreover, their local weights are reported in the last two columns of Tables 4-16, using the EM and model (8).
e local weights of three suppliers given each subcriterion of each criterion are presented in Tables 17-22 to obtain a compound weight vector of each criterion. In Table  23, the compound weight vectors are accumulated in the form of a comprehensive weight vector relative to the supplier accumulation. In Tables 17-23, the weights of EM and model (8)
According to Saaty [2], EM calculates eigenvector weights by solving the following eigenvector equations: n j�1 a ij w j � λ max w i , i � 1, . . . , n, where λ max is the maximum eigenvalue of pairwise comparison matrix A, w i (i � 1, . . . , n) applies in w i > 0, and we have n i�1 w i � 1. A weakness of EM is its non-linearity.
Should the matrix have greater dimensions, it will be timeconsuming to calculate eigenvalues and eigenvectors. Saaty and Vargas [35] formulated LSM as follows [37]: Model (11) is clearly non-linear. Chu et al. [10] formulated WLSM as follows: Model (12) is also non-linear. Crawford [11] formulated LLSM as follows: By solving Model (13), which is non-linear too, w i (i � 1, . . . , n) will be computed. By an overview, it can be stated that Ordinary Least Squares (OLS) and LLSM minimize arithmetic mean of errors and geometric mean of errors respectively.
Wang et al. [36] formulated CCMA as follows: where w * i is calculated by the following equations: where a j � 1/n n i�1 a ij .   Table 3: Results from the cross-weight evaluation based on DEA/AR models (6) and (8) Cost Analysis (C 13 )

Number of Employees (C 21 )
Organizational Structure (C 22 5 1.7471 (7) 0.0310 (7) 0.0465 (6) 0.0413 (7) 0.0356 (7) 0.0426 (7) w 6 2.0066 (6) 0.0363 (6) In this pairwise comparison matrix, we have β � 13.2372 Table 24 shows priority vectors obtained from different prioritization methods. As can be seen, various prioritization methods generate different rankings. is is mainly due to the fact that the initialization vector A was highly inconsistent and should be therefore adjusted. Lipovetsky and Conklin [37] showed that the order of the adjusted ranking is asw 8 > w 7 > w 1 > w 3 > w 2 > w 6 > w 5 > w 4 . Ranking in DEA/ AR Model (8) is exactly of the same order as ranking in EM.

Conclusion
Although obtaining weights in the AHP has already been investigated by different studies, this problem needs more clarification.
is paper proposed a new approach for generating weights from a pairwise comparison matrix. e proposed approach considers the most undesirable weight for each decision criterion or alternative and selects a decision criterion or alternative with the greatest weight. Two numerical examples, including BEKO in Turkey on CRT supplier selection, conducted by Sevkli, Lenny Koh [34], were investigated using the proposed approach. Moreover, the proposed method proved its ability to effectively differentiate decision criteria and alternative of DEAHP efficient.

Data Availability
All of our data are clear, and we have explained their source.

Conflicts of Interest
e authors declare that they have no conflicts of interest.