A New Linear Programming Method for Weights Generation and Group Decision Making

5/25/2018 A New Linear Programming Method for Weights Generation and Group Decision M...

http:///reader/full/a-new-linear-programming-method-for-weights-generation-and-group-

Group Decis Negot (2012) 21:233254DOI 10.1007/s10726-009-9182-x

A New Linear Programming Method for Weights

Generation and Group Decision Making in the AnalyticHierarchy Process

Seyed Saeed Hosseinian Hamidreza Navidi

Abas Hajfathaliha

Published online: 4 November 2009 Springer Science+Business Media B.V. 2009

Abstract This paper proposes a new linear programming method entitled by

LP-GW-AHP for weights generation in analytic hierarchy process (AHP) which

employs concepts from data envelopment analysis. We propose four specially con-

structed linear programming (LP) models which are used to derive weight vector from

a pair-wise comparison matrix or a group of them. We can use both interval and relative

importance weights for each decision maker in LP-GW-AHP. In this method, solving

only one LP model is enough for local weights derivation from pair-wise comparisonmatrices. Five numerical examples are examined to illustrate the potential applica-

tions of the LP-GW-AHP method. We show that not only derived weights of the new

method have slight differences than Saatys eigenvector weights but sometimes they

are better than eigenvector method weights in the fitting performance index as well.

LP-GW-AHP is compared with a method which has been recently proposed for the

weights derivation in the group AHP and it becomes obvious that LP-GW-AHP pro-

vides better weights. The simple additive weighting method is utilized to aggregate

local weights without the need to normalize them.

Keywords Group decision making Analytic hierarchy process Data

envelopment analysis Interval importance weight Relative importance weight

Fitting performance

S. S. Hosseinian (B) A. Hajfathaliha

Department of Industrial Engineering, Shahed University, P.O. Box 18155/159, Tehran, Iran

e-mail: [email protected]; [email protected]

A. Hajfathaliha

e-mail: [email protected]



j

234 S. S. Hosseinian et al.

1 Introduction

The analytic hierarchy process (AHP) was developed bySaaty(1980). AHP is one the

most popular multi criteria decision making MCDM tools for formulating and analyz-

ing decisions (Ramanathan 2006). Most of the MCDM techniques require numerousparameters, which are difficult to be determined precisely requiring extensive sen-

sitivity analysis (Srinivasa Raju and Nagesh Kumar 2006). How to deriving priority

vector from a pair-wise comparison matrix has being an important research topic in the

AHP and is substantially investigated in the AHP literature. 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 pro-

gramming method (FPM), the gradient eigenweight method (GEM), and so on (Wang

and Chin2009).Ho(2008), andVaidya and Kumar(2006) provided an overview ofAHP applications.

DEA is a nonparametric approach which was developed byCharnes et al.(1978)

based on linear programming to evaluate relative efficiency of similar decision mak-

ing units (DMUs) and it utilizes multiple inputs to produce multiple outputs. Relative

efficiency is defined as the ratio of total weighted output to total weighted input. The

main limitation of DEA models is running a separate linear program for each DMU.

This will be computationally intensive when the number of DMUs is large (Srinivasa

Raju and Nagesh Kumar2006). Interested Readers can see the more details of the

DEA theory and applications inCharnes et al.(1994) andCooper et al.(2000).Recently, some papers have used DEA approach for weights derivation from pair-

wise comparison matrices in AHP which they will be surveyed in next section. In this

paper, we propose a new method to generate weights from a pair-wise comparison

matrix or a group of them in AHP which uses concepts from DEA. We can use both

relative importance weight and interval importance weight for defining the possible

weights of each decision maker. This new method is called A linear programming

method to generate weights in the AHP (LP-GW-AHP). In this method, solving only

one LP model is enough for local weights derivation from a pair-wise comparison

matrix or group of them and it doesnt need to normalize derived weight vector.The paper is organized as follow: After introduction, Sect.2begins with a brief

literature survey on the integrated DEA-AHP approach then reviews weights genera-

tion methods that used DEA approach. In Sect. 3, we propose LP-GW-AHP method to

derive local weights in AHP and two numerical examples are examined and analyzed.

In Sect. 4, LP-GW-AHP method is extended to handle the importance of each decision

makers opinions. The aggregation of the local weights by SAW is provided in Sect.

5.The paper is concluded in Sect.6.

2 Literature Review

2.1 A Literature Survey on the Integrated DEA-AHP Methodology
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-



2.1 A Literature Survey on the Integrated DEA AHP Methodology

A New Linear Programming Method for Weights Generation 235

Table 1 The combined AHP-DEA approach and its specific areas

Sr. no. Author/s (year) Specific areas Other tool/s used

1 Tseng et al. (2009) Business performance evaluation TOPSIS

2 Korpela et al. (2007) Selecting the warehouse

operator network

3 Yang and Kuo (2003) Facility layout selection

problem

4 Farzipoor Saen et al. (2005) Evaluation the efficiency of

research organizations

5 Wang et al. (2008) Bridge risk assessment SAW

6 Takamura and Tone (2003) Relocation of several

government agencies

Assurance region

and Delphi

procedure

7 Ertay et al. (2006) Facility layout selection

problem

8 Azadeh et al. (2008) Railway system improvement

and optimization

Simulation model

9 Bowen (1990) Site selection

10 Shang and Sueyoshi (1995) Flexible manufacturing

system selection

Simulation model

11 Jyoti et al. (2008) Evaluation the performance

of R&D organizations

with minimization of inputs and/or maximization of outputs as associated objectives(Ramanathan 2006). In this section we do not want to survey the link between MCDM

and DEA, we especially survey link between AHP and DEA. The utilization of inte-

grated DEA-AHP approach is not new and there have been some utilizations of this

approach. Some of these studies are summarized in Table1and are described in the

following paragraphs.

Shang and Sueyoshi(1995)used the integrated DEA-AHP approach for the flexi-

ble manufacturing system selection of a manufacturing organization. They used three

modules: an AHP, a simulation module and an accounting procedure. Both AHP and

simulation models were used to generate the necessary outputs for the DEA and anaccounting procedure was used to determine the DMU outputs.

Farzipoor Saen et al.(2005) proposed the combined DEA-AHP approach to deter-

mine the relative efficiency for slightly non-homogeneous DMUs and demonstrated

the application of proposed method for the efficiency evaluation of eighteen Iranian

research organizations.

Azadeh et al.(2008) applied combined DEA-AHP approach for railway system

improvement and optimization. First, computer simulation was used to modeling,

verifying and validating the system. Second, AHP methodology was used to determi-

nate the weight of any qualitative criteria (inputs or outputs). Finally, the DEA model

used to solve the multi objective model to identify the best alternative.

Jyoti et al. (2008) used the integrated DEA-AHP approach for the performance
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-



Jyoti et al.(2008) used the integrated DEA AHP approach for the performance


Takamura and Tone(2003) used combined DEA-AHP approach to relocate sev-

eral government agencies out of Tokyo. There were eighteen criteria for site selection

problem and AHP was adopted to obtain the relative importance of the criteria. The

assurance regions model of DEA was used to evaluate effectiveness of alternatives.

Delphi procedure was discussed, too.Tseng et al.(2009) applied the synthetic DEA-AHP approach to determine the rel-

ative business performance of high-tech manufacturing companies. AHP was applied

to determine the relative importance weights of all indicators and dimensions. The

performance scores of the qualitative indicators were obtained by using fuzzy set the-

ory and cost efficiency indicator score was obtained by using the DEA method. By

normalizing the companies performance scores, including the qualitative and quan-

titative indicators, they aggregated the weights of the evaluated indicators. Finally,

Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) approach

was used to rank the performance of the companies.Korpelaet al. (2007) applied thecombinedDEA-AHP approach toselectwarehouse

operator network. AHP was used to derive priorities of seven subcriteria and the DEA

outputs were formed by AHP analysis. Both direct and indirect costs were accounted

as inputs. Finally, DEA was used for analyzing the service/cost-effectiveness of the

warehouse operators.

Some papers have used this approach to solve the facility layout design problems.

Ertay et al.(2006) applied the combined DEA-AHP approach to the facility layout

design that was very similar to which was presented inYang and Kuo(2003).

In these papers, AHP was often used for alternatives evaluation with respect to qual-itative criteria and DEA for final ranking. In another kind of integrated DEA-AHP

applications, AHP has utilized for full ranking DMU used in DEA (Sinuany-Stern

et al.2000). Some studies have separately applied AHP and DEA for a given problem

and have compared and explored different results (Tone 1989;Bowen 1990).

2.2 Weights Generation Methods with DEA Approach in the AHP

In this section, we briefly review the papers that have used DEA for weights derivation

from pair-wise comparison matrices.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 comparisons. 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 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 usedthe SAW method to get final weight for each bridge structure.
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-




2.2.1 DEAHP Method

Ramanathan(2006) used DEA for the local weights derivation from pair-wise com-

parison matrices for alternatives in the AHP. He proposed a new method and called

it DEAHP. Let A =

ai j

nnbe a pair-wise comparison matrix with ai i =1 and

ai j =1/ai jand W=(w1, . . . , wn)T be its weight vector. The DEAHP hasnoutputs

and one dummy constant input. Based on the input-oriented CCR model (1), the alter-

natives 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. When the DEAHP is applied for perfectly consistent matrices, it

estimates weights correctly.Sevkli et al.(2007) used the DEAHP model in a supplier

selection problem.

Max w0 =

nj=1

a0jvj subject to

u1 =1,n

j=1

ai jvj u1 0, i =1, . . . , n,

u1; vj 0, j =1, . . . , n.

(1)

2.2.2 DEA/AR Method

Wang et al.(2008a) showed that the DEAHP method has some drawbacks and pre-

sented that the DEAHP may produce counterintuitive local weights for inconsistentpair-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 regions (AR) for weights deri-

vation in the AHP and finally SAW (Hwang and Yoon 1981) used to get final weights.

The DEA/AR model was proposed as Follow:

Max w0 =

n

j=1

a0jvj subject towi =n

j=1 ai jvj 1, i =1, . . . , n,wj/ vj wj/n, j =1, . . . , n.

(2)

In this model, subscript zero refers to the decision criterion or alternative under eval-

uation and w0should be let w1, . . . , wn , respectively, and solve the above DEA/AR

model to derive weight vector. The second constraints in model (2)were provided as

follow:

Before describing them, first a Lemma is introduced.

Lemma 1 Let A = ai j nn

be a nonnegative matrix with nonzero row sums r1, . . . ,

rn and maximal eigenvaluemax. Then
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-




Since the nonnegative matrix A and its transpose AT have the same maximal eigen-

value, the above inequality also holds for the transpose of A, i.e.

mini

1ci

nj=1

ai j cj max max

i

1ci

nj=1

ai j cj (4)

where c1, . . . , cn are the column sums of A.

Consider the following characteristic equation for the pair-wise comparison matrix

A=

ai j

nn:

n

j=1

ai jwj =maxwi , i =1, . . . , n, (5)

max is the maximal eigenvalue of A. Eq. (5)can be rewritten as

nj=1

ai j (wj/max)= wi , i =1, . . . , n. (6)

Let

vj = wj

max, j =1, . . . , n, (7)

Then, Eq. (7) can be equivalently expressed as

nj=1

ai jvj =wi , i =1, . . . , n. (8)

For any pair-wise comparison matrix, it has already been known thatmax

n.

By Lemma1, the upper bound ofmax can also be determined. Let be the upper

bound formaxdetermined by Lemma1.Then we have n max . Using Eq.(7),

n max can be equivalently shown as n wj/vj for j =1, . . . , n that is

wj/ vj wj/n, j =1, . . . , n. (9)

These constraints were called assurance regions (AR).

2.2.3 Wang and Chin Method

Recently, Wang and Chin (2009) proposed a new DEA method for the weight vector
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-



Recently,Wang and Chin(2009) proposed a new DEA method for the weight vector


Table 2 The proposed DEA

view of the pair-wise

comparison matrix in the

LP-GW-AHP method

Criteria (alternative) Outputs with DEA view

1 2 n

Alternatives with DEA view

1 a11 a12 a1n

2 a21 a22 a2n

.

.

....

.

.

. ...

n an1 an2 ann

the group AHP situation. In these models, Ak =

aki j

nn

is a pair-wise comparison

matrix provided by the kth decision maker (DMk) (k=1, . . . , m) , hk>0 is its rel-

ative importance weight. The weights were calculated for each alternative or criterionsingly by using a separate model. This is called Wang and Chin method in this paper.

Max w0 =

nj=1

a0jzj subject to

nj=1

n

i=1

ai j

zj =1,

nj=1

ai jzj n zi , i =1, . . . , n

zj 0, j =1, . . . , n.

Max w0 =

nj=1

mk=1

hka(k)oj

zj subject to (10)

nj=1

(m

k=1

ni=1

hka(k)i j )zj =1,

nj=1

(m

k=1

hka(k)i j )zj n zi , i =1, . . . , n,

zj 0, j =1, . . . , n.

(11)

3 The LP-GW-AHP Method

In this section, we propose LP-GW-AHP method to derive local weights in AHP. In

DEA models, DMUs are placed in rows and outputs (inputs) in columns and the pur-

pose is either to minimize inputs or to maximize outputs. Now, Let A = (ai j )nnbe a pair-wise comparison matrix which is provided by a decision maker (DM) with

aii = 1, a

j i = 1/a

i j for j = i and W = (w

1, . . . , w

n)T be its weight vector.

With DEA approach, thei th criteria (alternative) of the matrix Ais viewed as a DMU

for(i = 1, . . . , n). Thus, the LP-GW-AHP method will have n DMUs. Also, j th cri-
http://-/?-http://-/?-http://-/?-http://-/?-


http:///reader/full/a-new-linear-programming-method-for-weights-generation-and-group

for(i 1, . . . , n). Thus, the LP GW AHP method will havenDMUs. Also, j th cri


We proposemodel (12) for the local weights derivation from a pair-wise comparison

matrix.

Max Z, Subject to

wi Z, i =1, . . . , n,

n

j=1

ai jvj wi =0, i =1, . . . , n,

ni=1

wi =1,

vi 1wi 0, i =1, . . . , n,

vi 1nwi 0, i =1, . . . , n,

wi 0; vi 0, i =1, . . . , n.

(12)

Here,wi (i =1, . . . , n)are the local weights for criteria (alternatives),vj (j =1, . . . ,

n) are the outputs weights which are determined by LP model and ai j (i, j =1, . . . , n)

are elements of pair-wise comparison matrix. In this LP model, the constraints vi

(1/)wi 0 and vi (1/n)wi 0 for (i =1, . . . , n)are assurance regions (AR)

which were described in Sect.2.2.2andis determined by Lemma1.That is

=min

max

i

1

ri

nj=1

ai j rj

,max

i

1

ci

nj=1

ai j cj

, (13)

where r1, . . . , rnand c1, . . . , cn are respectively the row sums and column sums ofthe pair-wise comparison matrix A =(ai j )nn .

Theorem1deals with the results of Model (12)for perfectly consistent comparison

matrices.

Theorem 1 model(12) produces true local priorities for perfectly consistent pair-

wise comparison matrices.

Proof let A = (ai j )nnbe a perfectly consistent pair-wise comparison matrix. Then

it can be characterized by a normalized priority vector W

= (w

1, . . . , w

n)

T

as A =(wi/wj )nn , where the priority vectorW

=(w1, . . . , wn)

Tsatisfiesn

i=1wi =1.

We should show thatwi =wi.

The fifth constraint of model (12) will be binded for any perfectly consistent

pair-wise comparison matrix, therefore vi =wi/n. So using the second, third and fifth

constraint of model (12), we will haven

i=1wi =n

i=1

nj=1ai jvj =

ni=1

nj=1

wiwj

wjn = 1

n(n

i=1wi)(n

j=1wjwj

)=1. Sincen

i=1wi =1 we will have

nj=1

wjwj

= n. Accordingly, the first constraint of model (12) can be written as

wi =nj=1

ai jvj =nj=1

wi

wj

wj

n

= wi

nn

j=1

wj

wj

= wi

n

n w

i

which completes the

proof
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-




Table 3 The DEA view of the pair-wise comparison matrices in the group AHP


1 2 N


1 k

mk=1

ak11

k

mk=1

ak12

k

mk=1

ak1n

2 k

mk=1

ak21

k

mk=1

ak22

k

mk=1

ak2n

.

.

....

.

.

. ...

n k m

k=1

akn1k

m

k=1

akn2 k

m

k=1

aknn

aggregated comparison matrix of a group of DMs. The geometric mean of some pair-

wise comparison matrices preserves thereciprocity. Therefore, we usegeometricmean

for aggregate pair-wise comparison matrices. Let Ak =

aki j

nn

be a pair-wise com-

parison matrix provided by the D Mk(k=1, . . . , m). In the group AHP situation, we

use k

mk=1a

ki j instead ofai j in model (12) to derive local weights from pair-wise com-

parison matrices. Where k

mk=1aki j is the geometric mean ofaki j for(k=1, . . . , m).The proposed DEA view of pair-wise comparison matrices in the group AHP situation

is shown in Table3.

Therefore, in the group AHP situation the LP model(12) is changed as follow:

Max Z, Subject to

wi Z, i =1, . . . , n,

nj=1

( k

mk=1

aki j )vj wi =0, i =1, . . . , n,

ni=1

wi =1,

vi 1wi 0, i =1, . . . , n,

vi 1nwi 0, i =1, . . . , n,

wi 0; vi 0, i =1, . . . , n.

(14)

In these LP models, Z is maximized under the condition that the same weights (vj )

are used in evaluating all DMUs (criteria or alternatives). One of the features of This

LP model is that it will be used only once for the weight vector derivation from a

pair-wise comparison matrix or the group of pair-wise comparison matrices in AHP.

Example 1 Consider the following pair-wise comparison matrices, in which A is bor-
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


http:///reader/full/a-new-linear-programming-method-for-weights-generation-and-group-d

Example 1 Consider the following pair wise comparison matrices, in which Ais bor


A=

1 1 4 5

1 1 5 3

1/4 1/5 1 3

1/5 1/3 1/3 1

, B =

1 2 5

1/2 1 3

1/5 1/3 1

, C=

1 9 5

1/9 1 3

1/5 1/3 1

,

D =

1 5 51/5 1 3

1/5 1/3 1

, E=

1 1 3 4 1

1 1 1 1/2 1/3

1/3 1 1 1/2 1/2

1/4 2 2 1 1/2

1 3 2 2 1

, F=

1 2 1 1 1

1/2 1 1/2 1 1

1 2 1 1/2 1

1 1 2 1 1

1 1 1 1 1

.

Accordingly, model (12) for the pair-wise comparison matrix Acan be written as

follow:

Max Z,

Subject to

w1 Z,

w2 Z,

w3 Z,

w4 Z,

v1+ v2+ 4v3+ 5v4 w1 =0,

v1+ v2+ 5v3+ 3v4 w2 =0,

(1/4)v1+ (1/5)v2+ v3+ 3v4 w3 =0,

(1/5)v1+ (1/3)v2+ (1/3)v3+ v4 w4 =0,

w1+ w2+ w3+ w4 =1,v1 (1/4.885)w1 0,

v2 (1/4.885)w2 0,

v3 (1/4.885)w3 0,

v4 (1/4.885)w4 0,

v1 (1/4)w1 0,

v2 (1/4)w2 0,

v3 (1/4)w3 0,

v4 (1/4)w4 0,

wi 0; vi 0, i =1, . . . , n.

As it was mentioned beforeis determined by Lemma1as bellow:

=min

maxi

1

ri

nj=1

ai j rj

,max

i

1

ci

nj=1

ai j cj

,

Therefore, for matrix A,calculation is shown in Table4.

The local weights for matrix Aare shown in Table5.Besides the LP-GW-AHP

method, we derive local weights by EM, DEAHP, DEA/AR and Wang and Chin
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-



method, we derive local weights by EM, DEAHP, DEA/AR and Wang and Chin


Table 4 Thecalculation for matrix Ain Example1

Row ri

nj=1

ai j rj1ri

nj=1

ai j rj Column ci

nj=1

ai j cj1ci

nj=1

ai j cj

1 11 48.133 4.376 1 2.45 9.967 4.068

2 10 48.850 4.885 2 2.533 11.050 4.362

3 4.45 14.800 3.326 3 10.333 36.800 3.561

4 1.867 8.883 4.759 4 12 62.850 5.238

=min {max(4.376, 4.885, 3.326, 4.759),max(4.068, 4.362, 3.561, 5.238}=4.885

from a pair-wise comparison matrix such as A =

ai j

nn

we need to solve nLP

models, but in LP-GW-AHP method, solving only one LP model is sufficient. Eigen-

vector method is nonlinear in nature (Wang and Chin 2009) but LP-GW-AHP methodis formulated as an LP model where it is much easier than EM to derive weights in

AHP.

As can be seen from Table5,derived weights from LP-GW-AHP method are not

too far from Saatys eigenvector weights and ranks from two methods are equal in all

matrices.

Finally, for comparing the quality of the derived local weights in different meth-

ods, we use fitting performance (FP) index, in which is measured by the following

Euclidean distance (Wang et al. 2008a):

FP=

1n2

ni=1

nj=1

(ai j wi/wj )2, ai j A,B,C,D,E,F. (15)

Since the elements of a consistent comparison matrix can be written as ratios of local

weights, its FP value will be zero. 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.

(15)wiandwjare the derived weights of each matrix. As can be seen from Table6,

derived weights of LP-GW-AHP method for the matrixCand Aperform better thanother methods in FPindex and are better than Saatys eigenvector weights for the

matrices Band D. Also LP-GW-AHP method weights of the pair-wise comparison

matricesEand Fhave slight differences from Saatys eigenvector method in fitting

performances.

Example 2 Consider four matrices A(1),A(2),A(3)andA(4) that are comparisons

about the relative importance of five alternatives with respect to a criterion. This

example is related to group AHP situation and there are four DMs in it. Thus, the

amount ofkis four in the model (14).

The LP-GW-AHP and the EM are used for weights derivation and are compared with
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-



The LP GW AHP and the EM are used for weights derivation and are compared with


Table 5 Local weights of thepair-wise comparison matrices (A,B,C,D,E,F), produced by different meth-

ods

Comparison matrix w1 w2 w3 w4 w5

A: Eigenvector method(EM)weightsA 0.400 0.393 0.128 0.078

B 0.582 0.309 0.109

C 0.764 0.149 0.087

D 0.701 0.202 0.097

E 0.314 0.135 0.1050160 0.285

F 0.223 0.152 0.200 0.232 0.193

B: DEAAHP method weights(normalized)

A 0.341 0.341 0.205 0.114

B 0.556 0.333 0.111

C 0.556 0.333 0.111

D 0.556 0.333 0.111

E 0.230 0.230 0.115 0.197 0.230

F 0.200 0.200 0.200 0.200 0.200

C: DEA/AR method weights(normalized)

A 0.396 0.395 0.129 0.080

B 0.582 0.309 0.109

C 0.747 0.159 0.093

D 0.695 0.205 0.099

E 0.306 0.138 0.106 0.162 0.288

F 0.223 0.153 0.201 0.230 0.193

D: Wang and Chin method weights(normalized)

A 0.396 0.394 0.129 0.080

B 0.582 0.309 0.109

C 0.754 0.155 0.091

D 0.705 0.205 0.099

E 0.315 0.136 0.106 0.161 0.282

F 0.222 0.153 0.201 0.232 0.193

E: LP-GW-AHP method weights

A 4.885 0.401 0.387 0.130 0.081

B 3.056 0.582 0.309 0.109

C 3.978 0.751 0.156 0.093

D 3.606 0.695 0.205 0.099

E 5.823 0.310 0.133 0.107 0.163 0.286

F 5.300 0.222 0.154 0.199 0.231 0.193




Table 6 Fitting performances by different local weights

Comparison

matrix

Consistency

ratio

Fitting performances by

EM DEAAHP DEA/AR Wang and Chin LP-GW-AHP

A 0.09 0.815 1.184 0.810 0.807 0.801

B 0.003 0.128 0.1160 0.128 0.127 0.127

C 0.31 1.854 2.450 1.804 1.813 1.792

D 0.12 0.950 1.119 0.912 0.929 0.913

E 0.09 0.596 0.843 0.599 0.595 0.597

F 0.04 0.306 0.387 0.307 0.307 0.307

Table 7 The local weights and

ranking of the alternatives by

EM and LP-GW-AHP methods

Alternative EM Rank LP-GW-AHP Rank

A1 0.3111 1 0.3102 1

A2 0.1738 4 0.1743 4

A3 0.2498 2 0.2506 2

A4 0.1797 3 0.1792 3

A5 0.0855 5 0.0858 5

FP 0.312 0.310

In Eq. (15), wiand wjare the derived weights of each method and ai jis the geometric

mean ofa ki j for(k=1, . . . , 4).

The LP-GW-AHP method solves one LP model to derive local weights from four

pair-wise comparison matrices that is too easier than eigenvector method.

A(1) =

1 1 3 4 2

1 1 1/2 2 5

1/3 2 1 3 6

1/4 1/2 1/3 1 1

1/2 1/5 1/6 1 1

A(2) =

1 2 3 1/9 2

1/2 1 1/8 1 1/2

1/3 8 1 1/2 4

9 1 2 1 5

1/2 2 1/4 1/5 1

(3)

1 3 1/5 7 51/3 1 1 1/3 2

(4)

1 2 3 5 21/2 1 1 4 5
http://-/?-http://-/?-http://-/?-



(3)

(4)


4 The LP-GW-AHP Method with Considering the Importance of Decision

Makers Opinions

4.1 Relative Importance Weights

Let Ak =

aki j

nn

be a pair-wise comparison matrix provided by the D Mk (k =

1, . . . , m), andhkbe its relative importance weight that satisfyingm

k=1hk=1. We

use them

k=1(aki j )

hkinstead of km

k=1aki j in model (14), that is the geometric mean

ofa ki jwith consideringh kfor(k=1, . . . , m).

Thus, the model (14)changes as follow:

MaxZ, Subject to

wi Z, i =1, . . . , n,n

j=1

(m

k=1

(aki j )hk)vj wi =0, i =1, . . . , n,

ni=1

wi =1,

vi 1wi 0, i =1, . . . , n,

vi 1nwi 0, i =1, . . . , n,

wi 0; vi 0, i =1, . . . , n.

(16)

This LP model is used to derive local weights when there are relative importanceweights for DMs.

Example 3 Consider fourpair-wise comparison matricesA(1),A(2),A(3)andA(4) that

are borrowed fromWang and Chin(2009). They are pair-wise comparison matrices

about the relative importance of five decision criteria in the group AHP. Four matrices

are provided by four DMs and each DM has relative importance weight(hk). Four

matrices are as follow:

A(1) =

1 1 3 4 1

1 1 1 1/2 1/3

1/3 1 1 1/2 1/2

1/4 2 2 1 1/2

1 3 2 2 1

A(2) =

1 8 1 2 2

1/8 1 1/8 1/3 1/5

1 8 1 2 1

1/2 3 1/2 1 1

1/2 5 1/2 1 1

A(3) =

1 8 1 1 1

1/8 1 1/8 1/5 1/8

1 8 1 2 1

1 5 1/2 1 1

1 8 1 1 1

A(4) =

1 2 1 1 1

1/2 1 1/2 1 1

1 2 1 1/2 1

1 1 2 1 1

1 1 1 1 1

This problem is solved by Wang and Chin method (model (10) in this paper), EM
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-



This problem is solved by Wang and Chin method (model(10) in this paper), EM


Table 8 Fitting performances by different local weights, produced by different methods in the group AHP

Sample Relative importance

weights of four DMs

Method Local weights

w1 w2 w3 w4 w5 FP

1 (50%, 30%, 15%, 5%) EM 0.3108 0.0793 0.1881 0.1724 0.2494 0.339

Wang and Chin 0.3091 0.0906 0.2054 0.1662 0.2287 0.378

LP-GW-AHP 0.3105 0.0795 0.1880 0.1724 0.2495 0.338

2 (50%, 16.7%, 16.7%, 16.7%) EM 0.3006 0.0913 0.1771 0.1810 0.2500 0.283

Wang and Chin 0.2967 0.1006 0.1912 0.1762 0.2353 0.300

LP-GW-AHP 0.2995 0.0911 0.1762 0.1798 0.2534 0.279

3 (25%, 25%, 25%, 25%) EM 0.2830 0.0749 0.2211 0.1891 0.2320 0.145

Wang and Chin 0.2803 0.0863 0.2291 0.1845 0.2198 0.229

LP-GW-AHP 0.2830 0.0749 0.2211 0.1890 0.2320 0.145

geometric mean ofa ki j for(k=1, . . . , 4)with considering DMs relative importance

weights.

As can be seen from Table8derived weights of LP-GW-AHP method are closer to

EM weights and perform better FP values rather than Wang and Chin method.

4.2 Interval Importance Weights

Let Ak =

aki j

nn

be a pair-wise comparison matrix provided by D Mk(k=1, . . . ,

m),andh kbe its interval importance weight, that hkkwithk, k[0, 1].

The proposed DEA view of pair-wise comparison matrices when there are interval

importance weights is shown in Table9.

We propose model (17) to derive local weights from pair-wise comparison matrices

when there are interval importance weights. This model can be equivalently expressedas model (19)linear form.

Max Z, Subject to

nj=1

hkaki jvj Z, i =1, . . . , n; k=1, . . . , m,

mk=1

nj=1

hkaki jvj wi =0, i =1, . . . , n,

ni=1

wi =1.

wi 0; vj 0,

i j 1

(17)
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-



i j 1 n


Table 9 The DEA view of the pair-wise comparison matrices when there are interval importance weights


1 2 n


DM1 1 a111

a112

a11n

2 a121

a122

a12n

.

.

....

.

.

. ...

n a1n1 a1n2

a1nn

.

.

....

.

.

....

.

.

....

DMm 1 am11

am12

am1n

2 am21 am22 a

m2n

.

.

....

.

.

. ...

n amn1

amn2 a

mnn

Ifkhkk,Then it can be equivalently expressed as

kh kkkskj

vjk sk j kvj 0, j =1, . . . , n,

sk j kvj 0, j =1, . . . , n. (18)

Then, model (17)is changed as follow:

Max Z, Subject to

nj=1

sk j aki j Z, k=1, . . . , m; i =1, . . . , n,

mk=1

nj=1

sk j aki j wi =0, i =1, . . . , n,

ni=1

wi=1.

sk j kvj 0, j =1, . . . , n,

sk j kvj 0, j =1, . . . , n,

wi 0; vj 0;skj 0, i, j =1, . . . , n; k=1, . . . , m.

(19)

when there are interval importance weights for DMs, model (19) is used to derive local

weights from a group of pair-wise comparison matrices in AHP.

Example 4 Consider four pair-wise comparison matrices A(1),A(2),A(3)andA(4) in

Example3and their interval importance weights for DMs are as follow:

0 1 h1 0 3
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-



0.1 h1 0.3,


Table 10 The local weights with interval importance weights for Example4

Local weights

w1 w2 w3 w4 w5

0.275 0.065 0.263 0.166 0.230

Fig. 1 A hierarchical structure by analytic hierarchy process

This example is solved by model(19) and the results are shown in Table10.

5 Aggregation of Local Weights to Get Final Weights

In this section, we propose the aggregation of local weights to get final weights by LP-

GW-AHP. In the DEAHP, DEA/AR and Wang and Chin models the sum of resulted

weights may be more than one because they are not normalized. However, it is neces-

sary to normalize them before aggregation to have the AHP final weights but when we

use LP-GW-AHP method to derive local weights there is no need to normalize local

weights before aggregation. This is one of the superiority of our model comparing

with other models discussed in this paper. A hierarchical structure in AHP is shown

in Fig.1that has mcriteria and nalternatives. Let w1, . . . , wmbe the local weights

ofmcriteria that all have been derived by LP-GW-AHP method and w1j , . . . , wn jbethe local weights for alternatives with respect to the j t h criterion(j =1, . . . , m). The

final weights are shown in the last column of Table11.

Example 5 In this example, we illustrate the SAW method using a hierarchical struc-

ture and data set that are taken from Chen and Huang (2004). The objective is selection

of high-tech industries for the Hsinchu Science-based Industrial Park (HSIP) where

placed at the top level of the analytic hierarchy is shown in Fig.2.Here, there are

seven criteria including utility consumption, technology support, industry relevance,

land supply, technology level, government policy, and market potential for assessing

objective. Finally, six high-tech industries are placed at the bottom of the analytic

hierarchy. They are: semiconductor, computer, communications, photo electronics,
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-



hierarchy. They are: semiconductor, computer, communications, photo electronics,


Table 11 Aggregation of local weights to get final weights

Alternative Criteria Final weights

w1 w2 wm

A1 w11 w12 w1m

mj=1

w1jwj

A2 w21 w22 w2m

mj=1

w2jwj

.

.

....

.

.

....

.

.

....

An wn1 wn2 wnmm

j=1

wn jwj

Semiconductor Computer Communications Photo-

electronics

Precision

equipment

Biotechnology

Utility

consumption

Technology

support

Industry

relevance

Land supply Technology

level

Government

policy

Market

potential

Selection of high-tech industries

Fig. 2 A hierarchical structure for selection of high-tech industries

means of the DMs pair-wise comparison matrices was calculated. For example, the

geometric mean of the pair-wise comparison matrices of criteria with respect to the

goal is shown in Table 12. We use LP-GW-AHP method to derive criteria local weights

and the results show in the last column of Table12. The alternatives local weights that

derived by LP-GW-AHP method are shown in Table13.The SAW method is used to

calculate alternatives final weights that are shown in the last column of Table13.

6 Concluding Remarks

In this paper, we proposed a new method to derive weights from a pair-wise com-

parison matrix or a group of pair-wise comparison matrices in AHP which employs

the concepts from DEA and called it LP-GW-AHP. Several numerical examples were

provided to illustrate the potential applications of LP-GW-AHP method.

The LP-GW-AHP method with utilizing the DEA approach has formulated as an
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-



The LP GW AHP method with utilizing the DEA approach has formulated as an


Pair-wisecomparisonsofcriteriawithrespecttothegoal

Utility

consumption

Technolo

gy

support

Industry

relevance

Landsupply

Technologylevel

Government

policy

Market

potential

Localweight

sumptio

n

1.

000

0.

746

0.

707

1.

046

0.

493

0.

456

0.

488

0.

092

ysupport

1.

000

0.

923

1.

561

0.

802

0.

794

0.

542

0.

129

levance

1.

000

1.

188

0.

804

0.

624

0.

523

0.

123

y

1.

000

0.

511

0.

611

0.

545

0.

097

ylevel

1.

000

1.

119

0.

642

0.

169

ntpolicy

1.

000

0.

697

0.

169

ential

1.

000

0.

222



P s y l ly y n e


Thefinalweightsforhigh-techindustries

Criteria

Utility

consumption

(0.0

92)

Technology

support

(0.1

29)

Industry

relevance

(0.1

23)

Lands

upply

(0.0

97)

Technology

level(0.1

69)

Government

policy(0.1

69)

Market

potential

(0.2

22)

Finalweight

ctor

0.

149

0.

138

0.

154

0.

181

0.

161

0.145

0.

173

0.

158

0.

118

0.

110

0.

132

0.

133

0.

127

0.088

0.

097

0.

112

ations

0.

158

0.

146

0.

157

0.

161

0.

142

0.157

0.

175

0.

157

tronics

0.

178

0.

173

0.

209

0.

183

0.

178

0.193

0.

224

0.

194

quipme

nt

0.

146

0.

210

0.

176

0.

160

0.

188

0.150

0.

135

0.

165

ogy

0.

252

0.

223

0.

173

0.

183

0.

204

0.267

0.

196

0.

213



T uc a tr q o


In this paper, we proposed an LP model for weights derivation from a pair-wise

comparison matrix and also some LP models in the group AHP situation. We showed

that the LP-GW-AHP method provides better fitting performance (FP) value than

Wang and Chin method in group AHP with relative importance weights for DMs.

In LP-GW-AHP method, we have needed to solve only one LP model to derivelocal weights from a pair-wise comparison matrix or several pair-wise comparison

matrices in group AHP, but in DEAHP, DEA/AR and Wang and Chin methods for

a pair-wise comparison matrix such as A =

ai j

nnwe need to solve nLP models.

In addition, we showed that LP-GW-AHP method is easy to use and has implemented

without the need to any normalization on derived weights.

These good characteristics make LP-GW-AHP method very practical and effective.

Further researches can extend the LP-GW-AHP method to handle fuzzy AHP and

improve the LP-GW-AHP method until it has fewer fitting performances of derived

local weights.

References

Azadeh A, Ghaderi SF, Izadbakhsh H (2008) Integration of DEA and AHP with computer simulation for

railway system improvement and optimization. Appl Math Comput 195:775785. doi:10.1016/j.amc.

2007.05.023

Bowen WM (1990) Subjective judgements and dataenvelopmentanalysis in site selection. Comput Environ

Urban 14(2):133144. doi:10.1016/0198-9715(90)90018-O

Charnes A, Cooper WW, Lewin AY, Seiford LM (1994) Data envelopment analysis: theory methodologyand application. Kluwer, Boston

Charnes A, Cooper WW, Rhodes E (1978) Measuring the efficiency of decision making units. Eur J Oper

Res 2:429444. doi:10.1016/0377-2217(79)90229-7

Chen CJ, Huang CC (2004) A multiple criteria evaluation of high-tech industries for the science-based

industrial park in Taiwan. Inform Manag 41:839851. doi:10.1016/j.im.2003.02.002

Cooper WW, Seiford LM, Tone K (2000) Data envelopment analysis: a comprehensive text with models,

applications, references and DEA-solver software. Kluwer Academic Publishers, Boston

Ertay T, Ruan D, Tuzkaya UR (2006) Integrating data envelopment analysis and analytic hierarchy for the

facility layout design in manufacturing systems. Inform Sci 176:237262. doi:10.1016/j.ins.2004.12.

001

Farzipoor Saen R, Memariani A, Hosseinzadeh Lotfi F (2005) Determining relative efficiency of slightly

non-homogeneous decision making units by data envelopment analysis: a case study in IROST. Appl

Math Comput 165:313328. doi:10.1016/j.amc.2004.04.050

Ho W (2008) Integrated analytic hierarchy process and its applicationa literature review. Eur J Oper Res

186:211228. doi:10.1016/j.ejor.2007.01.004

Hwang CL, Yoon K (1981) Multiple attribute decision making: methods and applications. Springer, Berlin

Jyoti T, Banwet DK, Deshmukh SG (2008) Evaluating performance of national R&D organizations

using integrated DEA-AHP technique. Int J Prod Perform Manag 57(5):370388. doi:10.1108/

17410400810881836

Korpela J, Lehmusvaara A, Nisonen J (2007) Warehouse operator selection by combining AHP and DEA

methodologies. Int J Prod Econ 108:135142. doi:10.1016/j.ijpe.2006.12.046

Ramanathan R (2006) Data envelopment analysis for weight derivation and aggregation in the analytic

hierarchy process. Comput Oper Res 33:12891307. doi:10.1016/j.cor.2004.09.020Saaty TL (1980) The analytic hierarchy process. McGraw-Hill, New York

Sevkli M, Lenny Koh SC, Zaim S, Demirbag M, Tatoglu E (2007) An application of data envelopment
http://dx.doi.org/10.1016/j.amc.2007.05.023http://dx.doi.org/10.1016/j.amc.2007.05.023http://dx.doi.org/10.1016/j.amc.2007.05.023http://dx.doi.org/10.1016/0198-9715(90)90018-Ohttp://dx.doi.org/10.1016/0198-9715(90)90018-Ohttp://dx.doi.org/10.1016/0377-2217(79)90229-7http://dx.doi.org/10.1016/0377-2217(79)90229-7http://dx.doi.org/10.1016/j.im.2003.02.002http://dx.doi.org/10.1016/j.ins.2004.12.001http://dx.doi.org/10.1016/j.ins.2004.12.001http://dx.doi.org/10.1016/j.ins.2004.12.001http://dx.doi.org/10.1016/j.amc.2004.04.050http://dx.doi.org/10.1016/j.amc.2004.04.050http://dx.doi.org/10.1016/j.ejor.2007.01.004http://dx.doi.org/10.1016/j.ejor.2007.01.004http://dx.doi.org/10.1108/17410400810881836http://dx.doi.org/10.1108/17410400810881836http://dx.doi.org/10.1108/17410400810881836http://dx.doi.org/10.1016/j.ijpe.2006.12.046http://dx.doi.org/10.1016/j.ijpe.2006.12.046http://dx.doi.org/10.1016/j.cor.2004.09.020http://dx.doi.org/10.1016/j.cor.2004.09.020http://dx.doi.org/10.1016/j.cor.2004.09.020http://dx.doi.org/10.1016/j.ijpe.2006.12.046http://dx.doi.org/10.1108/17410400810881836http://dx.doi.org/10.1108/17410400810881836http://dx.doi.org/10.1016/j.ejor.2007.01.004http://dx.doi.org/10.1016/j.amc.2004.04.050http://dx.doi.org/10.1016/j.ins.2004.12.001http://dx.doi.org/10.1016/j.ins.2004.12.001http://dx.doi.org/10.1016/j.im.2003.02.002http://dx.doi.org/10.1016/0377-2217(79)90229-7http://dx.doi.org/10.1016/0198-9715(90)90018-Ohttp://dx.doi.org/10.1016/j.amc.2007.05.023http://dx.doi.org/10.1016/j.amc.2007.05.023



y g g pp p


Sinuany-Stern Z, Abraham M, Yossi H (2000) An AHP/DEA methodology for ranking decision making

units. Int T Oper Res 7(2):109124. doi:10.1016/S0969-6016(00)00013-7

Srinivasa RajuK, Nagesh Kumar D (2006) Ranking irrigation planning alternatives usingdataenvelopment

analysis. Water resour Manag 20:553566. doi:10.1007/s11269-006-3090-5

Takamura Y, Tone K (2003) A comparative site evaluation study for relocating Japanese government agen-

cies out of Tokyo. Socio Econ Plan Sci 37:85102. doi:10.1016/S0038-0121(02)00049-6Tone K (1989) A comparative study on AHP and DEA. Int J Poly Inform 13:5763

Tseng FM, Chiu YJ, Chen JS (2009) Measuring business performance in the high-tech manufacturing

industry: a case study of Taiwans large-sized TFT-LCT panel companies. Omega-Int j Manag S 37:

686697. doi:10.1016/j.omega.2007.07.004

Vaidya OS, Kumar S (2006) Analytic hierarchy process: an overview of applications. Eur J Oper Res 169:

129. doi:10.1016/j.ejor.2004.04.028

Wang YM, Chin KS (2009) A new data envelopment analysis method for priority determination and group

decision making in the analytic hierarchy process. Eur J Oper Res 195:239250. doi:10.1016/j.ejor.

2008.01.049

Wang YM, Chin KS, Yang JB (2007) Three new models for preference voting and aggregation. J Oper Res

Soc 58:13891393Wang YM, Chin KS, Poon GKK (2008a) A data envelopment analysis method with assurance region for

weight generation in the analytic hierarchy process. Decis Support Syst 45: 913921. doi:10.1016/j.

dss.2008.03.002

Wang YM, Liu J, Elhag TMS (2008b) An integrated AHP-DEA methodology for bridge risk assessment.

Comput Ind Eng 54:513525. doi:10.1016/j.cie.2007.09.002

Yang T, Kuo C (2003) A hierarchical DEA-AHP methodology for the facilities layout design problem. Eur

J Oper Res 147:128136. doi:10.1016/S0377-2217(02)00251-5
http://dx.doi.org/10.1016/S0969-6016(00)00013-7http://dx.doi.org/10.1016/S0969-6016(00)00013-7http://dx.doi.org/10.1007/s11269-006-3090-5http://dx.doi.org/10.1007/s11269-006-3090-5http://dx.doi.org/10.1016/S0038-0121(02)00049-6http://dx.doi.org/10.1016/S0038-0121(02)00049-6http://dx.doi.org/10.1016/j.omega.2007.07.004http://dx.doi.org/10.1016/j.ejor.2004.04.028http://dx.doi.org/10.1016/j.ejor.2008.01.049http://dx.doi.org/10.1016/j.ejor.2008.01.049http://dx.doi.org/10.1016/j.ejor.2008.01.049http://dx.doi.org/10.1016/j.dss.2008.03.002http://dx.doi.org/10.1016/j.dss.2008.03.002http://dx.doi.org/10.1016/j.dss.2008.03.002http://dx.doi.org/10.1016/j.cie.2007.09.002http://dx.doi.org/10.1016/j.cie.2007.09.002http://dx.doi.org/10.1016/S0377-2217(02)00251-5http://dx.doi.org/10.1016/S0377-2217(02)00251-5http://dx.doi.org/10.1016/S0377-2217(02)00251-5http://dx.doi.org/10.1016/j.cie.2007.09.002http://dx.doi.org/10.1016/j.dss.2008.03.002http://dx.doi.org/10.1016/j.dss.2008.03.002http://dx.doi.org/10.1016/j.ejor.2008.01.049http://dx.doi.org/10.1016/j.ejor.2008.01.049http://dx.doi.org/10.1016/j.ejor.2004.04.028http://dx.doi.org/10.1016/j.omega.2007.07.004http://dx.doi.org/10.1016/S0038-0121(02)00049-6http://dx.doi.org/10.1007/s11269-006-3090-5http://dx.doi.org/10.1016/S0969-6016(00)00013-7



Reproducedwithpermissionof thecopyrightowner. Further reproductionprohibitedwithoutpermissio

A New Linear Programming Method for Weights Generation and Group Decision Making

Documents

Transcript of A New Linear Programming Method for Weights Generation and Group Decision Making