A New Method Based on Data Envelopment Analysis to Derive Weight Vector in the Group Analytic Hierarchy Process

INTRODUCTION

Analytic Hierarchy Process (AHP), since invention has been a tool at the hands of decision makers and researchers; it is one of the most widely used Multiple Criteria Decision Making (MCDM) tools (Vaidya and Kumar, 2006). The AHP is used in almost all application related to MCDM in the last 20 years (Ho, 2008). How to weight vector derivation from pair-wise comparison matrices has being an important research topic in the AHP. Apart from Saaty’s well-known Eigenvector Method (EM), quite a number of alternative approaches have been suggested such as the Weighted Least-Square Method (WLSM), the Logarithmic Least Squares Method (LLSM), the Geometric Least Squares Method (GLSM), the Fuzzy Programming Method (FPM), the gradient method (GEM) and so on (Wang and Chin, 2009). The utilization of integrated DEA-AHP approach is not new and there have been some utilizations of this approach. For example, Shang and Sueyoshi (1995) used it for the selection of flexible manufacturing system, Saen et al. (2005) applied this approach for efficiency evaluation of 18 Iranian research organizations, Azadeh et al. (2008) used combined DEA-AHP for railway system improvement and Jyoti Banwet and Deshmukh (2008) used the integrated DEA-AHP for the performance evaluation of Indian research and development organizations. Some studies have used this approach to solve the facility layout design problems (Yang and Kuo, 2003; Ertey et al., 2006). In these studies, AHP was often used for the evaluation of alternatives with respect to qualitative criteria and DEA for final ranking. In another kind of integrated DEA-AHP applications, AHP has been used for full ranking the DMUs used in DEA (Sinuany-Stern et al., 2000).

Most of the Multi Criteria Decision Making (MCDM) techniques require numerous parameters, which are difficult to be determined precisely requiring extensive sensitivity analysis. On the other hand, the main limitation in DEA is that standard formulation of DEA creates a separate linear program for each DMU. This will be computationally intensive when the number of DMUs is large (Raju and Kumar, 2006). Some studies have used DEA approach for weights derivation from pair-wise comparison matrices in AHP. Ramanathan (2006) has proposed DEAHP method. Wang et al. (2008a) has proposed a DEA model with Assurance Region (AR) for priority derivation in AHP. Recently, Wang and Chin (2009) have proposed two LP models for weight vector derivation in the AHP. Their models need to solve separate LP models for each criterion (alternative). Also, derived weights are not normalized and it only uses the relative importance weight for each decision maker.

In this study, we propose a new method for the weight vector derivation of the pair-wise comparison matrices for alternatives or criteria in the group AHP situation, which uses the concept of variable weights of DEA. We called this method as the method based on the DEA for the weight vector derivation in the group decision making (DEA-WDGD). Using this method, solving only one LP model is enough for the local weights derivation of several pair-wise comparison matrices in the group decision making. Also, the obtained weights are normal and it is not necessary to be normalized again. We can use both the relative importance weight and the interval importance weight for identifying the possible weights for each decision maker.

Analytic hierarchy process: Analytic Hierarchy Process (AHP) was developed by Saaty (1980). The AHP is one the most popular MCDM tools for formulating and analyzing decisions (Ramanathan, 2006). The strength of the AHP lies in its ability to arrange complex multi-person and multi-attribute problems hierarchically and then to investigate each level of the hierarchy separately, combining the results as the analysis progresses (Franklin-Liu and Hai, 2005). The AHP was adopted in personal, social, manufacturing sector, political, engineering, education, industry, government, sports and management (Vadiya and Kumar, 2006).

Some key and basic steps involved in this methodology are as follow (Vadiya and Kumar, 2006):

Data envelopment analysis: Data Envelopment Analysis (DEA) is a nonparametric approach which is developed by Charnes et al. (1978) based on linear programming to evaluate relative efficiency of similar Decision Making Units (DMUs) and utilize multiple inputs to produce multiple outputs. The DEA application for assessing efficiency includes three stages (Golany and Roll, 1989): the first stage is to identify appropriate DMUs. Then, the inputs and outputs must be selected for the measurement of relative efficiency of DMUs. Finally, DEA model is applied to analyze the data.

Assume that there are N DMUs producing J outputs using I inputs. Let the mth DMU produce outputs y_mj using x_mi inputs. The resulting output-input structure of DMUs is shown in Table 1. The objective of the DEA models is to identify the DMU that produces the highest amounts of outputs by consuming lowest inputs. The efficiencies of other inefficient DMUs are obtained relative to the efficient DMUs and are assigned efficiencies score between zero and one. The efficiency scores are computed using mathematical programming (Ramanathan, 2006).

Weights derivation methods from pair-wise comparison matrices by using DEA approach: Here, we go over the papers that have used DEA for weights derivation from pair-wise comparison matrices.

Ramanathan (2006) used DEA for the local weights derivation from pair-wise comparison matrices for alternatives in the AHP. He proposed a new method and called it DEAHP. Let A = (a_ij)_nxn be a pair-wise comparison matrix with a_ii = 1 and a_ij = 1/a_ij and W = (w₁,…,w_n)^T be its weight vector. The DEAHP has n outputs and one dummy constant input. Based on the input-oriented CCR model 1, the alternatives weights were calculated separately for each alternative using a separate linear programming model. This method was used for the aggregation of the local weights to get final weights. Using DEAHP for consistent matrices leads to estimating correct weights. Mehmet et al. (2007) used the DEAHP method in a supplier selection problem.

Table 1:	DMUs, inputs and outputs for a DEA model
Ramanathan (2006)

(1)

Wang et al. (2008a) showed that the DEAHP method has some drawbacks and presented that the DEAHP may produce counterintuitive local weights for inconsistent pair-wise comparison matrices and the DEAHP is sometimes over insensitive to some comparisons in a pair-wise comparison matrix. To overcome these drawbacks of the DEAHP, They proposed a DEA method with Assurance Region (AR) for weights derivation in the AHP and finally SAW is used to get final weights.

Recently, Wang and Chin (2009) proposed a new DEA method for the weight vector derivation from a pair-wise comparison matrix, extended it to the group AHP situation and utilized the SAW to get final weights. They proposed model 2 for the weight vector derivation from a pair-wise comparison matrix and proposed model 3 in the group AHP situation. In these models, is a pair-wise comparison matrix provided by the k^th Decision Makers (DM_k) (k = 1,…,m) and h_k>0 is its relative importance weight. In model 2 and 3, the weights were calculated for each alternative or criterion by using a separate model.

(2)

(3)

In some studies, linguistic terms and ordinal numbers have used in the decision making to rank DMUs and this has not been based on the pair-wise comparison matrix. For example, Wang et al. (2007) proposed two LP models and a nonlinear programming (NLP) model to assess weights and utilized ordinal numbers to rank DMUs. Wang et al. (2008b) proposed an integrated DEA-AHP methodology to evaluate bridge risks of hundreds or thousands of bridge structures, based on the maintenance priorities. They utilized AHP only to determine the criteria weights. Linguistic terms such as high, medium, low and none were utilized to assess bridge risks under each criterion and DEA methodology to determine the value of the linguistic terms. Finally, they used the SAW method to get final weight for each bridge structure.

THE DEA-WDGD METHOD

Using the concept of DEA in DEA-WDGD method: In DEA models, DMUs lay in the rows and outputs and inputs indices lay in columns of datasheet table. The purpose is either to maximize outputs or to minimize inputs. Now, Let be a pair-wise comparison matrix which is provided by the DM_k with a_ii = 1, a_ij = 1/a_ij for j ≠ i and (k = 1,…m). The concept of the new DEA method is shown in Table 2. In Table 2, ith row (i = 1,…,n) shows the ith criterion or alternative which is viewed as a DMU. Thus, the DEA-WDGD method will have n DMUs. Elements of A^k are the pair-wise comparisons of the DM_k and the sum of its ith row is shown in the kth column and ith row of Table 2 for (k = 1,..,m; i = 1,..,n). Each column of Table 2 is viewed as an output. Thus, the DEA-WDGD method will have m outputs.

We propose the following model for the local weights derivation from the pair-wise comparison matrices used in the group AHP:

(4)

where, w_i(i = 1,…,n) are the local weights of criteria or alternatives, v_k is the output weight of DM_k which is determined by model 4 for (k = 1,..,m) and (i, j = 1,…,n; k = 1,…,m) are elements of pair-wise comparison matrices. Since, the objective function maximizes the sum of weights, it reaches to its maximum value of the optimal solution.

Table 2:	The proposed DEA view of pair-wise comparison matrices in the group AHP
Outputs with DEA view

So, the resulted weights will be normalized. In this LP model, Z is maximized under the condition that the same weights are used in evaluating all DMUs, this principle is in accordance of DEA. One of the features of This LP model is that it will be used only once for the weight vector derivation from pair-wise comparison matrices in the group AHP.

The DEA-WDGD method with considering the importance of decision makers’ opinions
Relative importance weights for Dms: Let be a pair-wise comparison matrix provided by the DM_k (k = 1…,m) and h_k be its relative importance weight that satisfying:

Thus:

(5)

We can not use the Eq. 5 in model 4 because v_k is variable in model 4. So, Eq. 5 are expressed equivalently as:

(6)

Now, we can replace the linear constraints (6) instead of constraints (v₁–v_k = 0, k = 2,…,m) of model 4.

Interval importance weight for Dms: Let be a pair-wise comparison matrix provided by the DM_k (k = i,…,m) and h_k be its interval importance weight that α_k≤h_k≤ β_k with α_k, β_k ∈ [0, 1]. With considering the first and second constraints of the model 4,

and will be less than 1. Thus, h_k can be expressed equivalently as:

in model 4 which is normalized weight importance. Therefore:

Let

If α_k≤h_k≤ β_k, then it can be expressed equivalently as:

(7)

Now, we should replace the linear constraints (7) instead of constraints (v₁–v_k = 0, k = 2,…,m) of model 4.

NUMERICAL EXAMPLES

We provide three numerical examples to illustrate the potential applications of the DEA-WDGD in the group AHP situation. The first example compares the DEA-WDGD with Wang and Chin (2009) method and shows that derived weights of the DEA-WDGD are better than it. The second one compares derived weights from DEA-WDGD with Saaty’s eigenvector method. The third example utilizes the interval importance weights for DMs.

Example 1: Consider four pair-wise comparison matrices about the relative importance of five decision criteria in the group AHP, which are borrowed from Wang and Chin (2009). Four DMs are provided by four DMs which are shown as follow. Each DM has relative importance weight (h_k).

This example is solved by DEA-WDGD method and Wang and Chin method (model 3 in this study). The derived weights by these methods are shown in Table 3. In order to compare the quality of local weights derived by different methods, we use Fitting Performance (FP) index, which is measured by the following Euclidean distance (Wang et al., 2008a):

(8)

The FP value of a consistent comparison matrix is zero because its elements can be written as ratios of local weights.

Table 3:	The local weights produced by different methods for example 1

Table 4:	The derived local weights and the ranks for alternatives by different methods

Table 5:	The local weights and output weights for example 3

For inconsistent matrices, the FP value will not be zero but the smaller value of FP shows the higher quality of the derived weights. In Eq. 8, w_i and w_j are the derived weights of each method and E is the geometric mean of pair-wise comparison matrices.

As for model 4 and Eq. 6 the LP model for the sample 1 of Table 3 can be written as follow:

As can be shown From Table 3, the weights derived from DEA-WDGD have better FP values comparing with Wang and chin’s model.

Example 2: Consider four pair-wise comparison matrices A⁽¹⁾, A⁽²⁾, A⁽³⁾, A⁽⁴⁾ that are comparisons about the relative importance of five alternatives with respect to a criterion. The DEA-WDGD and the EM are used to derive weights and a comparison between the results is provided. In the EM, the geometric mean is used to aggregate four matrices.

As can be shown from Table 4, two methods have equal ranks and the derived weights are not too far from each other. Eigenvector method is a nonlinear method but the DEA-WDGD method is formulated as an LP model that is much easier to derive weights than EM.

Example 3: Consider four pair-wise comparison matrices A⁽¹⁾, A⁽²⁾, A⁽³⁾, A⁽⁴⁾ in example 1 and suppose Interval importance weights for DMs as follow:

Before solving model 4 to derive weights we should replace the following constraints instead of constraints (v₁–v_k = 0, k = 2,…,m) of model 4. The results are shown in Table 5.

AGGREGATION OF LOCAL WEIGHTS TO GET FINAL WEIGHTS

Here, we propose the aggregation of local weights to get final weights by DEA-WDGD. In the Wang and Chin (2009) model the sum of resulted weights may be more than one because they are not normalized. However, it is necessary to normalize them before aggregation to have the AHP final weights but when we use DEA-WDGD to derive local weights, there is no need to normalize local weights before aggregation and this is one of the superiority of present model. A hierarchical structure in AHP is shown in Fig. 1 that has m criteria and n alternatives. Let (w₁,…,w_m) be the local weights of m criteria that all have been derived by DEA-WDGD and w_1j,…,w_nj be the local weights for m alternatives with respect to the jth criterion (j = 1,…,m). The final weights are shown in the last column of Table 6.