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 Saatys 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
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):
||State the problem
||Broaden the objectives of the problem or consider all actors,
objectives and its outcome
||Identify the criteria that influence the behavior
||Structure the problem in a hierarchy of different levels constituting
goal, criteria, sub-criteria and alternatives
||Compare each element in the corresponding level and calibrate
them in the numerical scale. This requires n (n1)/2 comparisons, where
n is the No. of elements with the considerations that diagonal elements
are equal or 1 and the other elements will simply be the reciprocals of
the earlier comparisons
||Perform calculations to find the maximum Eigen-value, Consistency
Index (CI), Consistency Ratio (CR) and normalized values for each criterion/alternative
||If the maximum Eigen-value, CI and CR are satisfied then decision
is taken based on the normalized values; else the procedure is repeated
till these values lie in a desired range
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 ymj using xmi 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 = (aij)nxn be
a pair-wise comparison matrix with aii = 1 and aij = 1/aij
and W = (w1,
,wn)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.
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
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 kth Decision Makers
(DMk) (k = 1,
,m) and hk>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.
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 DMk with
aii = 1, aij = 1/aij for j ≠ i and (k =
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 Ak are the pair-wise comparisons of the DMk
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:
where, wi(i = 1,
,n) are the local weights of criteria or alternatives,
vk is the output weight of DMk 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.
||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
Relative importance weights for Dms: Let
be a pair-wise comparison matrix provided by the DMk (k = 1
and hk be its relative importance weight that satisfying:
We can not use the Eq. 5 in model 4 because vk
is variable in model 4. So, Eq. 5 are expressed equivalently
Now, we can replace the linear constraints (6) instead of constraints (v1vk = 0, k = 2,
,m) of model 4.
Interval importance weight for Dms: Let
be a pair-wise comparison matrix provided by the DMk (k = i,
and hk be its interval importance weight that αk≤hk≤ βk
with αk, βk ∈ [0, 1]. With considering
the first and second constraints of the model 4,
and will be less than 1. Thus, hk can be expressed equivalently as:
in model 4 which is normalized weight importance. Therefore:
If αk≤hk≤ βk, then it can be
expressed equivalently as:
Now, we should replace the linear constraints (7) instead of constraints (v1vk = 0, k = 2,
,m) of model 4.
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 Saatys 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 (hk).
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):
The FP value of a consistent comparison matrix is zero because its elements
can be written as ratios of local weights.
|| The local weights produced by different methods for example
|| The derived local weights and the ranks for alternatives
by different methods
|| 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, wi and wj 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 chins 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 (v1vk = 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,
be the local weights of m criteria that all have been derived by DEA-WDGD and
,wnj 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.
In this study, we proposed the DEA-WDGD method to derive weight vector in the
group AHP situation by utilizing DEA approach. Some numerical examples were
provided and we showed that the derived weights of DEA-WDGD are better than
Wang and Chin (2009) model. After comparing the derived
weights of DEA-WDGD with eigenvector method it was shown that the derived weights
of DEA-WDGD are acceptable. The new DEA-WDGD method has the following good characteristics:
||The DEA-WDGD with utilizing the DEA approach is formulated
as an LP model and is much easier than eigenvector method to derive weights
in the group AHP
||In the DEA-WDGD, solving one LP model is sufficient for the
local weights derivation of several pair-wise comparison matrices in the
group AHP situation
||The DEA-WDGD method Utilize the simple additive weighting
method for aggregation of local weights without the need to normalization
||The DEA-WDGD can use both the interval importance weight and
the relative importance weight for each decision maker
||A hierarchical structure by analytic hierarchy process
|| Aggregation of local weights to get final weights
Further researches can extend the DEA-WDGD to handle fuzzy AHP and enable it to derive weights from only one pair-wise comparison matrix.