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Information Systems 32 (2007) 1005–1017

A model for selecting an ERP system based on linguistic

information processing

Xiuwu Liao, Yuan Li



, Bing Lu

Management School of Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

Abstract

An enterprise resource planning (ERP) is an enterprise-wide application software package that integrates all necessary
business functions into a single system with a common database. In order to implement an ERP project successfully in an
organization, it is necessary to select a suitable ERP system. This paper presents a new model, which is based on linguistic
information processing, for dealing with such a problem. In the study, a similarity degree based algorithm is proposed to
aggregate the objective information about ERP systems from some external professional organizations, which may be
expressed by different linguistic term sets. The consistency and inconsistency indices are defined by considering the subject
information obtained from internal interviews with ERP vendors, and then a linear programming model is established for
selecting the most suitable ERP system. Finally, a numerical example is given to demonstrate the application of the
proposed method.

r2006 Elsevier B.V. All rights reserved.

Keywords: ERP system; Information systems; Linguistic modeling; Information processing

1. Introduction

In today’s dynamic and unpredictable business
environment, companies face the tremendous
chal-lenge of expanding markets and rising customer
expectations. This compels them to lower total costs

in the entire supply chain, shorten throughput
times, reduce inventories, expand product choice,
provide more reliable delivery dates and better
customer service, improve quality, and efficiently
coordinate globe demand, supply and production

[1,2]. In order to accomplish these objectives, more
and more companies are turning to the enterprise
resource planning systems (ERP). An ERP is a
packaged enterprise-wide information system that
integrates all necessary business functions, such as
product planning, purchasing, inventory control,
sales, financial and human resources, into a single
system with a shared database[3,4].

A successfully implemented ERP can offer
organizations the following three major benefits

[5,6]:



Automating business process



Timely access to management information



Improving supply chain management through

the use of e-commerce

www.elsevier.com/locate/infosys

0306-4379/$ - see front matter r 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.is.2006.10.005

Corresponding author. Tel.: +86 029 82668953;
fax: +86 028 82668953.

E-mail addresses: (X. Liao),

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In the past few years, thousands of companies
around the world have implemented ERP systems.
The number of companies that plan to implement
ERP is growing rapidly. Since the early to
mid-1990s, the ERP software market has been and
continues to be one of the fastest growing segments
of the information technology (IT) industry [7].
AMR Research, an authoritative market forecast
institution in America, indicated that the ERP
market would grow at annual rate of 37% in recent
5 years. The sales of the ERP packaged software are
estimated to be around $20 billion by the year 2000
and the eventual market size is predicted to be
around $1 trillion by the year 2010 [8]. Even in
China, a developing country, ERP has also become
a main product in the software market and the sales
have approached 600 million RMB in the first half
of 2002 ([9–11]). Surprisingly, given the significant
investment in resources and time, many companies
did not achieve success in ERP implementation. It is
estimated that the failure rate of ERP
implementa-tion ranges from 40% to 60% or higher [2]. Some
surveys and researches indicate that successful

outcome is also not guaranteed even under ideal
circumstances. Researchers consider that the factors
such as organizational change and process
re-engineering, the enterprise-wide implications, the
high resource commitment, and high potential
business benefits and risks associated with ERP
systems make the implementation a much complex
exercise [12,13]. It is therefore not surprising that
numerous companies have abandoned their ERP
projects before completion or have failed to achieve
their business objectives after implementation[14].
Many experts and scholars have investigated this
issue from various angles. Some provide valuable
insights into ERP implementation process and
others identify a variety of factors that can be
considered to be critical to the success of an ERP
implementation. These factors include top
manage-ment support, business plan and vision,
organiza-tional change management and culture, business
process re-engineering (BPR), data accuracy,
educa-tion and training, and vendor seleceduca-tion and support,
etc. ([2,3,5,12,13,15–20]). A successful ERP project
involves managing business process change,
select-ing an ERP software system, implementselect-ing this
system, and examining the practicality of the
system. However, a wrong ERP system selection
would either fail the project or weaken the system to
an adverse impact on company performance. Due
to limitations in available resources, the complexity

of ERP systems, and the diversity of alternatives, it
is often difficult for an organization to select a
suitable ERP system[21].

The complexity of ERP system makes it difficult
for a single decision maker to consider all aspects of
problem. The organization which plans to
imple-ment ERP project usually employs multiple experts
from different sections in selection process. ERP
system selection, therefore, can be viewed a
multi-attribute group decision-making (MAGDM)
pro-blem. It involves multiple attributes, which are not
easy to quantify. So decision makers must deal with
vague or imprecise information in the evaluation
process of ERP system. A reasonable approach for
dealing with such a problem may be to use linguistic
assessment to represent the subjective judgment of
decision makers by means of linguistic variables,
that is, variables whose values are words or
sentences in a natural or artificial language [22,23].
Each linguistic value is characterized by a label and
a semantic value. The label is a word or sentence
belonging to a linguistic term set and semantic value
is a fuzzy subset in a universe of discourse[24]. After
Zadeh introduced fuzzy set theory to deal with
vague problems, linguistic variables have been used
in approximate reasoning within the framework of
fuzzy set theory to handle the ambiguity in
evaluating data and the vagueness of linguistic
expression. Thus, the fuzzy linguistic approach is

appropriate for some problems in which
informa-tion may be qualitative, or quantitative informainforma-tion
may not be stated precisely. So far, a number of
MAGDM approaches have been proposed for
dealing with linguistic assessment information in
literatures ([24–32]). These methods can be briefly
classified into the following three categories[33]

(1) The approximative computational model based
on the Extension Principle.

(2) The ordinal linguistic computational model.
(3) The 2-tuple linguistic computational model.

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expression domain. The models in the second
category is also called symbolic model. It makes
direct computations on labels using the ordinal
structure of the linguistic term sets. But symbolic
method easily results in a loss of information caused
by the use of the round operator. The models in last
one use the 2-tuple linguistic representation and
computational model to make linguistic
computa-tions. Research results [33] show such linguistic
information processing manner can effectively avoid
the loss and distortion of information. It has a
distinct advantage over other linguistic processing
methods in accuracy and reliability. At present, only
a few group decision-making approaches based on
the 2-tuple linguistic model are proposed in
literatures. For example, Herrera et al. [30]present

a group decision making process for managing
non-homogeneous information. The
non-homoge-neous information can be represented as values
belonging to domains with different nature as
linguistic, numerical and interval valued or can
be values assessed in label sets with different
granularity, multi-granular linguistic information.
Herrera-Viedma et al. [34] present a model of
consensus support system to assist the experts in
all phases of the consensus reaching process of
group decision-making problems with
multi-gran-ular linguistic preference relations. Wang and
Fan [31] propose a method for solving multiple
attribute group decision making problems with
linguistic assessment information. In this method,
the two-tuple linguistic representation model is
used to aggregate the linguistic assessment
informa-tion. The optimal alternative is determined by
calculating the linguistic distance of every
alter-native and positive ideal solution and negative ideal
solution.

At present, some methods have been proposed
and applied to ERP or other information system
(IS) selection. Buss[35]employed a ranking method
to compare computer projects. This method is too
simple to reflect opinions of decision maker.
Teltumbde[36]proposed a methodology framework
for evaluating ERP projects based on the NGT and
AHP. Lee and Kim[37]combined the ANP and 0-1

goal programming model to select an information
system. Wei and Wang [38] presents a
comprehen-sive framework for combining objective data
obtained from external professional reports and
subjective data obtained from internal interviews
with vendors to select a suitable ERP system. The
Extension Principle is used to aggregate the

linguistic evaluation descriptions. Wei et al. [21]

proposed an AHP-based approach to ERP system
selection. This study uses the analytical framework
of AHP to synthesize decision maker’ tangible and
intangible measures with respect to numerous
competing objective inherent in ERP system
selec-tion and facilitate the group decision-making
process.

This paper presents a new model, which is based
on the 2-tuple linguistic information processing, for
dealing with the problem of ERP system selection.
In the study, a similarity degree based algorithm is
proposed to aggregate the objective information
about ERP systems from some external professional
organizations, which may be expressed by different
linguistic term sets. The consistency and
inconsis-tency indices are defined by considering the subject
information obtained from internal interviews with
ERP vendors, and then a linear programming
model is established for selecting the most suitable

ERP system.

This paper is organized as follows. In Section 2,
we give a brief review of 2-tuple representational
model and computational model. In Section 3, we
outline the model of ERP system selection based on
linguistic information processing. In Section 4, an
application is presented to illustrate the whole
decision process. Finally, some concluding remarks
are pointed out.

2. The 2-tuple linguistic model

Let S ¼ {s0,s1,y,sg} be a linguistic term set with

granularity g+1. In general, the granularity of
S should be small enough so as not to impose
useless precise levels on users but big enough to
allow a discrimination of the assessments in a
limited number of degrees [34]. Furthermore, we
suppose that S satisfies the following some
char-acteristics:

(1) The set is ordered: siXsjif iXj.

(2) There is a negation: Neg(si) ¼ sgi

(3) There is the max operator: max{si,sj} ¼ si if

siXsj

(4) There is the min operator: min{si,sj} ¼ siif siXsj

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linguistic term set with granularity 7, denoted as S,
could be given as follows:

S ¼ {s0¼none (N), s1¼very low (VH), s2¼low

(L), s3¼medium (M), s4¼high (H), s5¼very high

(VH), s6¼perfect (P)}.

Theoretically, the universe of the discourse over
which the term set is defined can be arbitrary, but
usually, linguistic term sets are defined in the
interval [0,1]. We assume that the semantics of
labels are given by fuzzy members defined in the
[0,1] interval, which are described by triangular
membership functions. For example, we may assign
the semantics to the set of seven terms, which is
shown inFig. 1.

The 2-tuple fuzzy linguistic approach was first
introduced by Herrera [29,40] for overcoming the
drawback of the classical computational models,
which include the semantic model and symbolic
model. The main advantages of this formalism to
cope with linguistic information over classical
models are summarized as follows: (1) The linguistic
domain can be treated as continuous, whilst in the

classical models it is treated as discrete, (2) The
linguistic computational model based on linguistic
2-tuples carries out processes of ‘‘computing with
words’’ easily and without loss of information, (3)
The results of the processes of ‘‘computing with
words’’ are always expressed in the initial linguistic
domain[33].

The 2-tuple linguistic model is a kind of new
information processing method. It takes 2-tuple to
represent linguistic assessment information and
carry out operation. The basic concept of linguistic
2-tuple is symbolic translation.

Definition 1. (Herrera and Martı´nez [29]). Let
S ¼ {s0,s1,y,sg} be a linguistic term set and b be

the result of an aggregation of the indexes of a set of
labels assessed in S, i.e., the result of a symbolic
aggregation operation bA[0,g]. Let i ¼ round (b)
and a ¼ bi be two values, such that, iA[0,g] and

aA[0.5,0.5) then a is called a Symbolic
Transla-tion.

From Definition 1, we can see that symbolic
translation refers to a value that lies in interval
[0.5,0.5). It represents the difference between b
and the closest term si(i ¼ round (b)) in S.

Definition 2. (Herrera and Martı´nez [29]). Let
S ¼ {s0,s1,ysg} be a linguistic term set and

bA[0,g] a value representing the result of a symbolic
aggregation operation, then the 2-tuple that
ex-presses the equivalent information to b is obtained
with the following function:

D: ½0; g ! S  ½0:5; 0:5ị
Dbị ẳ si; aị

si; i ẳ roundbị;

a ẳ b  i; a 2 ẵ0:5; 0:5ị;
(

where round(.) is the usual round operation, siis the

closest index label to b and a is the value of
symbolic translation.

Proposition 1. (Herrera and Martı´nez [29]). Let
S ¼ {s0,s1,y,sg} be a linguistic term set and (si,ai) is

a 2-tuple. There is always a D1function, such that,
from a 2-tuple it returns its equivalent numerical value
bA[0,g].

Remark. From Definition 2 and Proposition 1, it is
obvious that the conversion of a linguistic term si

into a linguistic 2-tuple by adding a value 0 as
symbolic translation, i.e., siAS ) (s_i,0).

Based on above definition, we can easily give the
computational models of 2-tuple. These models
include comparison of 2-tuple, negation operator
and aggregation operator of 2-tuple [29].

(1) Comparison of 2-tuples: Let (si,ai) and (sj,bj) be

two 2-tuples defined in the same linguistic term
set:

If i4j, then (si,ai) is bigger than (sj,aj), i.e.

(si,ai)4(sj,aj),

If i ¼ j, then



If ai4aj, then (si,ai)4(sj,aj),



If aI¼aj, then (si,ai) and (sj,aj) represent the

same value, i.e. (si,ai) ¼ (sj,aj).



If aI¼aj, then (si,ai) is small than (sj,aj), i. e.

(si,ai)o(sj,aj).

(2) Negation operator of 2-tuples: This operator
is defined as follows: Negsi; aiịị ẳDg 
D1si; aiịịịwhere siAS ẳ {s₀,s₁,y,s_g}

(3) Aggregation of 2-tuple.

0 0.17 0.33 0.5 0.67 0.83 1

N VL L M H VH P

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Definition 3. (Herrera and Martı´nez [29]). Let
a ¼ {(b1,a1),(b2,a2),y,(bn,an)} be a set of linguistic

2-tuples, the 2-tuple arithmetic mean operator z1is

z1ẵb1; a1ị; b2; a2ị; . . . ; bn; anị

ẳD D

1_b
i; aiị
n

Xn
iẳ1

ẳD 1

n
Xn

iẳ1
b_i

!!
,
where biẳD

1

(bi,ai) ¼ i+aI.

Definition 4. (Herrera and Martı´nez [29]). Let
a ¼ {(b1,a1),(b2,a2),y,(bn,an)} be a set of linguistic

2-tuples and (w1,w2,y,wn) be their associated

weights. The 2-tuple weighted average operator z2is

z2ẵb1; a1ị; b2; a2ị; . . . ; bn; anị

ẳD

Pn

iẳ1D1bi; aiịwi
Pn

iẳ1wi

 

ẳD

Pn
iẳ1biwi
Pn

iẳ1wi

 

,
where biẳD1(bi,ai) ẳ i+aI.

Definition 5. Let a ¼ {(b1,a1),(b2,a2),y,(bn,an)} be a

set of linguistic 2-tuples and W ẳ fr1; a01ị;
r2; a02ị; . . . ; ðrn; a0nÞg be their associated linguistic
2-tuple weights, The 2-2-tuple linguistic weighted
average operator z3is

z3ẵb1; a1ị; r1; a10ịị; b2; a2ị; r2; a02ịị; . . . ,
bn; anị; rn; a0nịị

ẳD

Pn

iẳ1D1ri; a0iị D1bi; aiị

Pn

iẳ1D1ri; a0iị

!
.

Denition 6. Let a ẳ {(b1,a1),(b2,a2),y,(bn,an)} and

b ẳ {(c1,b1,),(c2,b2),y,(cn,bn)} be two vectors of

linguistic 2-tuple over term set S, and w ¼ ((r1,e1),

(r2,e2),y,(rn,en)) their associated linguistic 2-tuple

weights, then 2-tuple linguistic weighted Euclidean
distance between a and b is defined as follows:

da; bị ẳ

Xn

jẳ1

D1rj; jịẵD1bj; ajị D1cj; bjị2

Pn
jẳ1D

1_r
j; jị

v
u
u

t . (1)

Eq. (1) gives an effective and simple method for
calculating the distance between two vectors of
linguistic 2-tuple. From above expression, we can
obtain the following some results:

(1) Two vectors of linguistic 2-tuple a and b are
identical if and only if the distance measurement
d(a,b) ¼ 0.

(2) dða; bÞ ¼ jD1ðb1; a1Þ D1ðc1; b1Þj;
if a ¼ ðb1; a1Þ; b ¼ ðc1; b1Þ.
(3) dða; bÞp max

j¼1;2;;njD
1_ðb

j; ajÞ D1ðcj; bjÞj.

3. Model and approach

In order to present the ERP system selection

frame-work, a stepwise procedure is first described as follows:
Step 1: Form a project team and conduct the
business process re-engineering.

Step 2: Collect all possible information about
ERP vendors and systems. Filter out unqualified
vendors.

Step 3: Establish the attribute hierarchy (see[38]).
Step 4: Employ the external professional
organi-zations to give evaluation information of each ERP
system with respect with each attribute.

Step 5: Aggregate all external professional
eva-luation information to obtain an objective decision
matrix.

Step 6: Every member in project team interviews
vendors, examines the vendor’s demonstrations,
collects detailed information, and finally gives the
partial order relation of candidate ERP systems
based on his own subjective judgment and
knowl-edge.

Step 7: Combine the evaluations of both
informa-tion sources to make final selecinforma-tion.Fig. 2shows the
whole framework of the method.

Considering the following ERP selection
pro-blem: Suppose there exist m possible ERP systems

(or alternatives x1,x2,y,xm to be evaluated by K

professional organizations (or experts) e1,e2,y,eK

on n attributes c1,c2,y,cn. ek (k ¼ 1,2,y,K) is

expert ek’s weight, such as kX0; PKk¼1k¼1. It
represents the relative importance of ek. Suppose xkij
is the rating of alternative xi (i ¼ 1,2,y,m) on

attribute cj (j ¼ 1,2,y,m), which is represented by

the label in the linguistic term set Sjk selected by
expert ek (k ¼ 1,2,y,K), where Sjk¼ fsjk0; s

jk
1; . . . ;
sjk

gjkg. In evaluation process, each expert expresses

his/her preferences depending on the nature of the
alternatives and on his/her own knowledge over
them. Therefore, the linguistic term sets Sjk; j ¼
1; 2;. . . ; m; k ¼ 1; 2; . . . ; K may be different. The
objective evaluation with linguistic assessment
information from expert ek(k ¼ 1,2,y,K) can then

be concisely expressed in matrix format as follows:

Dk¼
xk

11 xk12    xk1n
xk

21 xk22    xk2n
..

.
..
.

..
.

..
.
xk

m1 xkm2    xkmn
2

6
6
6
6
6
4

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3.1. The unification of linguistic assessment
information

In evaluation process of ERP system, experts may
have different knowledge, background and
discri-mination ability. Thus, they may use different
linguistic terms to express their opinions. In this
context, the linguistic term sets Sjk (j ¼ 1,2,y,n,
k ¼ 1,2,y,K) may have a different granularity and/
or semantics. In order to manage such linguistic
assessment information, we must make it uniform,
i.e., the multi-granularity linguistic assessment
information provided by all decision makers must

be transformed into unified linguistic term set, i.e.,
basic linguistic term set (BLTS), represented by ST

[28]. Before defining a transformation function, we
have to decide how to choose the BLTS, ST. In

general, ST must be a linguistic term set which

allows us to maintain the uncertainty degree
associated to the ability of discrimination of
decision maker to express the performance values.
The principle of choosing a BLTS is described as
follows[30].

(1) When there is only one term set with the
maximum granularity in Sjk, j ¼ 1,2,y,n, k ¼

1,2,y,K, then, it is chosen as ST;

<i>Decision matrix D</i>₁

<i>Decision matrix D</i>₂
.

.
.

<i>Decision matrix D_K</i>

Preference relationΘ1

Preference relationΘ2
.

.
.

Preference relationΘ<i>H</i>

<i>e</i>₁

<i>e</i>₂

<i>e_K</i>

<i>Transfer D_k (k = 1, 2..., K ),</i>

into 2-tuple linguistic
<i>decision matrix D_k</i>

Calculate group
decision matrix

<i>D by means of </i>

Similarity degree
based aggregation
algorithm

Gather the preference
information of all
decision makers on
alternative pairs:

<i>H</i>
<i>k </i>=1Θ

<i>k</i>
Θ = U
Calculate the
importance degree of
alternative pairs in Θ

Rank the orders of ERP systems

Define the group consistency and inconsistency indices

Constructing the linear programming model for obtaining the
weights of attributes and the positive ideal solution

<i>m</i>₁

<i>m</i>₂

<i>m_H</i>

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If we have two or more linguistic term sets with
maximum granularity, then STis chosen depending

on the semantics of these linguistic term sets, finding
two possible situations to establish ST: (a) if all the

linguistic term sets have the same semantics, then ST

is any of them; (b) there are some linguistic term sets
with different semantics. Then, ST is a basic

linguistic term set with a larger number of terms
than the number of terms that a decision maker is
able to discriminate.

After BLTS is chosen, each linguistic assessment
term set Sjk (j ¼ 1,2,y,n; k ¼ 1,2,y,K) can be
transformed into a fuzzy set in ST by using the

following transformation function.

Definition 7. (Herrera et al. [30]). Let Sjk¼
fsjk₀; sjk₁; . . . ; sjk_g

jkg and ST ¼ fs0; s1; . . . ; sgg be two

linguistic term sets and gXg_jk, then a
multi-granularity transformation function t_Sjk_S

T is dened

as
t_Sjk_S

T : S

jk _!_{F S}
Tị
t_Sjk_S

Ts

jk

i ị ẳ fsl; aijk_l Þjl 2 f0; 1; . . . ; ggg; 8sjki 2Sjk
aijk_l ẳmax

y minfmsjki yị; mslyịg,

where F(ST) is the set of fuzzy sets defined in ST

m_sjk

iðyÞ and mslðyÞ are the membership functions

associated to the linguistic terms sjk_i and sl,

respectively.

Furthermore, the linguistic assessments expressed
by means of fuzzy set on the BLTS can be
transformed into linguistic 2-tuple over the ST. This

transformation is carried out by using the following
function w[30]:

w: F STị ! ẵ0:g,
wt_Sjk_S

Ts

jk

i ÞÞ ¼wðfðsl; aijkl Þ,
l ¼ 0; 1; . . . ; ggị ẳ D

Pg
lẳ0la

ijk
l

Pg

lẳ0a
ijk
l

!
.

Therefore, utilizing the functions t and w, all fuzzy
decision matrices Dk; k ¼ 1; 2; . . . ; K can be
trans-formed into the normalized decision matrix Dkẳ
xk

ijịnn, where xkij; i ẳ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n; k ¼
1; 2;. . . ; K are linguistic 2-tuples on BLTS ST.

For the sake of convenience, let xk

ijẳ skij; akijị, where
sk

ij2ST and akij2 0:5; 0:5. Dk can be written

explicitly as

Dkẳ
sk

11; ak11ị sk12; a12k ị    ðsk1n; ak1nÞ

ðsk

21; ak21Þ ðsk22; a22k Þ    ðsk2n; ak2nÞ
..

.

..
.

..
.

..
.
ðsk

m1; akm1Þ ðskm2; am2k Þ    ðskmn; akmnÞ
2

6
6
6
6
6
4

3
7
7

7
7
7
5
.

3.2. Similarity degree based objective information
aggregation

After decision matrix Dkẳ xkijịmn; k ¼ 1; 2; . . . ;
K is calculated, respectively. We give a similarity

degree based algorithm to integrate Dk,

k ¼ 1,2,y,K into objective decision matrix D ẳ
xijịịmn: The aggregation process is carried out in
following steps:

(1) Calculating the similarity degree simðxk_ij; xl_ijị of
the assessment values of alternative xi

(i ẳ 1,2y,m) with respect to attribute cj

(j ¼ 1,2,y,n) between decision makers ek and

el, 1pk, lpK, k6ẳl.

The value D1xk
ijị D

1_xl
ijị




  ẳ D1sk

ij; alijÞ




D1ðsk

ij; alijÞjcan be used to measure the distance
between xk

ij and xlij: Thus the similarity degree
simðxk_ij; xl_ijÞcan be defined as follows[34]:

sim xk_ij; xl_ijị ẳ1  D
1_xk

ijị D
1_xl

ijị
g











,

where g+1 is the granularity of BLTS ST. The

range of simðxk

ij; xlijÞis the closed interval [0, 1].
The closer simðxk

ij; xlijÞto 1 the more similar xkij
and xl

ij are; while the closer simðxkij; xlijÞto 0 the
more distant xk

ij and xlij are.

(2) Establishing the similarity matrix SMij of the

assessment values of alternative xi(i ¼ 1,2,y,m)

with respect to attribute cj(j ẳ 1,2,yn), where

SMijẳ ẵsim xkij; xlijịKK,
and simxk

ij; xlijị ẳ1; if k ¼ l. Therefore, the
diagonal elements of SMijare unity.

(3) Calculating the average similarity degree
SMij(ek) and relative similarity degree RSMij(ek)

of decision maker ek (k ¼ 1,2,y,k) on the

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with respect to attribute cj(j ẳ 1,2,yn), where

SMijekị ẳ
PK

lẳ1;laksim xkij; xlijị

K  1 ,

RSMijekị ẳ

SMijekị
PK

lẳ1SMijelị
.

(4) Calculating the importance degree bk_ijof decision
maker ek (k ¼ 1,2,y,K) in the aggregation of

the assessment values xl

ij; l ¼ 1; 2; . . . ; K; where
bk_ijẳ_P_KkRSMijekị

lẳ1ẵlRSMijekị
.

(5) At last, calculating the group decision matrix
D ẳ (xij))m  n by using operator z2 which has

been defined in Section 2, where
xijẳz2ẵs1ij; a1ijị; s2ij; a2ijị; . . . ; skij; akijị

ẳD

PK
lẳ1D

1_sl
ij; alijịb

l
ij
Pn

iẳ1b
l
ij

!

ẳD XK

lẳ1D
1_sl

ij; alijịblij

 

ẳ bij; aijị.

The similarity degree-based aggregation
algo-rithm, as described above, considers not only the
relative importance of expert, but considers the
similarity of opinions of experts. Therefore,
it can make aggregation results reflect the
collective opinions more reasonably and more
objective.

3.3. Determining the ranking order of alternatives
In this section, we give a new decision approach
based on the group consistency and inconsistency
indices to determine the ranking orders of all ERP
systems.

3.3.1. The importance degree of preference relations

Assume that there are H members mk,

k ¼ 1,2,y,H in the project team. pkis mk’s weight,

such as pkX0; SHk¼1pk¼1: It represents the relative
importance of mk. Support the preference relations

on alternatives provided by the member mk

(k ¼ 1,2,y,H) is

Yk¼ fðp; qÞjxpxq; p; q ¼ 1; 2; . . . ; mg,

where xpxqmeans that either member mkprefers

xkto xqor mkis indifferent between xpand xq. Let

Y ¼ [H
k¼1Y

k _{be the set of preference relations on}

alternatives provided by all members of project team.
For each pair of alternatives (p,q)AY, it corresponds
to preference relation xpxq. In general, such a
preference relation is either the opinion of a member,
or the opinions of several members. It also may be the

views of all members. In order to identify the
importance of preference relations, we dene

m_pqẳ X

p;qị2YK

p_k.

Obviously, mpqẳpk, if only member mk thinks

xpxq, mpq¼1, if all members think xpxq.
Especially, when all decision makers have the equal
weight in a decision activity, then

m_pqẳ#fmkjp; qị 2 Y
k_{; m}

k2Eg

H ,

where # represents the cardinality of set {ek|(p,q)AYk,

mkAE}, E ¼ k ¼ 1,2,y,H}. In this paper, m_pq is
called as the important degree of alternative pair
(p,q).

3.3.2. Group consistency and inconsistency indices
For convenience, Let xi¼(xi1,xi2,y,xin) ¼

((bi1,ai1),(bi2,ai2),y,(bin,an)) and the most preferred

alternative by all group members (i.e., positive ideal
solution) be x¼((b1,a1),(b2,a2),y,(bn,an)), where

bij; bj 2ST; aij2 ð0:5; 0:5; aj2 ð0:5; 0:5; i ¼ 1,2,
y_{,m; j ¼ 1,2,y,n. Let L ¼ {l}₀_{, l}₁_,y,l_h_{} be another}
linguistic term set for assessing the importance of
attributes and w ¼ ((r1,e1),(r2,e2),y,(rn,en)) is the

weight vector of attributes represented in linguistic
2-tuple form, where rjAL, e_jA[0.5,0.5), j ¼
1,2,y,n. When xand w have been determined,
according to Definition 6 in Section 2, the 2-tuple
linguistic weighted Euclidean distance between xi

and xcan be written as
Viẳdxi; xị

ẳ X

n

jẳ1
D1_r

j; jịẵD1bij; aijị D1bj; ajị2
Pn

jẳ1

D1rj; jÞ
0

B
B
B
@

1
C
C
C
A

1=2

.

ð2Þ
From Eq. (2), we can easily get that Vi belongs to

interval [0,g]. Furthermore Viẳ0, if xiẳx. Let

ojẳ

D1rj; jị
Pn

jẳ1D1rj; jị

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then ojX0; Snjẳ1ojẳ1 and bij; bj2 ẵ0; g. Eq. (2)
can be simplied as

Viẳ
Xn

jẳ1

ojbijbjị2
!1=2

.

For the sake of convenience, we use the square of Vi

to measure the distance between xi and positive

ideal solution x:
diẳV2i ẳ

Xn
jẳ1

ojbijbjị2; i ẳ 1; 2; . . . ; m.

If the weight vector w ¼ ((r1,e1),(r2,e2),y,(rn,en))

and the positive ideal solution x¼((b1,a1),

(b2,a2),y,(bn,an)) are chosen by the group already,

the square of the 2-tuple linguistic weighted
Euclidean distance between alternatives xp,xq and

the positive ideal solution is calculated as follows:
dpẳ

Xn
jẳ1

ojbpjbjị2, (3)

dqẳ
Xn

jẳ1

ojbqjbjị2, (4)

8p; qÞ 2 Y, the alternative xp is closer to the

positive ideal solution than xq, if dqXdp. So the

ranking of alternatives xpand xqdetermined by dp

and dpis consistent with the preference given by one

or several decision makers. Conversely, if dpXdq,

then the ranking of alternatives xp and xq

deter-mined by dp and dq is inconsistent with the

preference given by one or several decision makers.
It means that x and w are nor chosen properly.
Therefore, we define an index, called ðdqdpÞ, to
measure inconsistency between the ranking order of
alternatives xpand xqdetermined by dpand dqand

the preference given by one or several decision
makers as follows:

dqdpị
ẳ

0 dqXdp

mpqdpdqị dqodp
(

ẳmaxf0; mpqdpdqịg.
5ị
From Eq. (5), we easily see that the ranking of
alternatives xp and xq determined by dp and dq is

consistent with alternative pair (p,q), if dqXdp.

Hence, inconsistency degree ðdqdpÞ is defined

to be 0; on the other hand, if dpXdq, then the

ranking of alternatives xpand xqdetermined by dp

and dqis inconsistent with alternative pair (p,q). The

more the difference between dpand dqis, the higher

the inconsistency degree. Considering the important
degrees of alternative pair (p,q), ðdqdpÞ is
defined to be mpq(dpdq). Hence, an inconsistency

index of the group based on w and x can be
denoted as

B ẳ X

p;qị2Y

dqdpị. (6)

In a similar way, a consistency index of the group is
defined as

G ẳ X

p;qị2Y

dqdpịỵ, (7)

where
dqdpịỵ

ẳ

mpqdqdpị; dqXdp

0; dqodp

(

ẳmaxf0; mpqdqdpịg.

8ị
The consistency index G measures the consistent
degree between the rankings of alternative
deter-mined by distance model and the preference given
by decision makers. The bigger G is, the higher the
consistency degree.

From the definitions of (dqdp)


and (dqdp)+,

we easily obtain following equation:
dqdpịỵ dqdpịẳmpqdqdpị.

3.3.3. Construct linear programming model to
determine the ranking order of alternatives

In order to determine positive ideal solution x
and weight vector w, we construct the following
mathematical programming model:

min B ẳ X

p;qị2Y

maxf0; m_pqdpdqịg

s:t: G  BXh

ojX0 j ¼ 1; 2; . . . ; n
Xn

j¼1

oj ¼1,

0pbjpg j ¼ 1; 2; . . . ; n ð9Þ

where h is a non-negative number provided by the
project team.8ðp; qÞ 2 Y, let

lpq ẳmaxf0; mpqdpdqịg.
Then, we have

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Thus, mathematic programming problem (9) can be
transformed into

min B ẳ X

p;qị2Y
lpq
s:t: G  BXh

m_pqdpdqị lpqp0; p; qị 2 Y
ojX0; j ẳ 1; 2; . . . ; n

Xn
j¼1

oj¼1

0pb_jpg; j ¼ 1; 2; . . . ; n

lpqX0 ðp; qÞ 2 Y. ð10Þ

Using Eqs. (2)–(8) and supposing vj¼ojbj

(j ¼ 1,2,y,n), the linear programming problem
(10) can be rewritten as follows:

min B ẳ X

p;qị2Y
lpq

s:t: X

n

jẳ1
oj

X
p;qị2Y

m_pqb2_qjb2_pjị

" #

2X

n

jẳ1
vj

X
p:qị2Y

mpqbqjbpjị

" #

X_h

Xn

jẳ1

ojẵmpqb2pjb
2
qjị

2X

n

jẳ1

vjẵmpqbpjbqjị
lpqp0p; qị 2 Y
ojX0; j ẳ 1; 2; . . . ; n

Xn
j¼1

oj¼1

0pvjpgoj; j ¼ 1; 2; . . . ; n

lpqX0 ðp; qÞ 2 Y. ð11Þ

In (11), the constraint 0pvjpgoj, j ¼ 1,2,y,n is

obtained from vj¼ojbj and bjA[0,g], j ¼ 1,2,y,n.
By solving the above linear programming using the
Simplex method, we can obtain optimal solution

ðo

1; o2; . . . :on; v1; v2; . . . ; vnÞ.

Furthermore, we can get the positive ideal solution
using following equation:

x_{ẳ b}

1; a1ị; b2; a2ị; . . . ; bn; anịị

ẳ D n


1
o
1
 

; D n

2
o
2
 

; . . . ; D n

3
o

3
 

 

. 12ị

After the weight vector o _{ẳ o}

1; o2; . . . ; onÞ and
positive ideal solution x are determined from the
linear programming model (11), the distance
be-tween alternative xiand xcan be computed using

following equation:
diẳ

Xn
jẳ1

o_jẵD1bij; aijị D1bj; ajị2.

The ranking orders of all alternatives can be
obtained according the increasing order of di.

4. A numerical example

This section presents a numerical example to
illustrate the method proposed in this paper.
Suppose an organization plans to implement ERP

system. The first step is to form a project team
E ¼ {m1,m2,m3} that consists of CIO and two senior

representatives from user departments. By collecting
all possible information about ERP vendors and
systems, project term choose four potential ERP
systems x1,x2,x3,x4 as candidates. The company

employs three external professional organizations
(or experts) e1,e2,e3to aid this decision-making. The

Project team selects four criteria to evaluate the
alternatives: (1) function and technology c1, (2)

strategic fitness c2, (3) vendor’s ability c3; (4)

vendor’s reputation c4. c1,c2,c3,c4are unquantifiable

due to their nature. So the experts provide the
ratings of alternatives with respect to these
attri-butes by means of linguistic variables. The linguistic
term sets and associated semantics of labels used
here are given in Table 1. We shall use the model
proposed in this paper to solve this problem.
(1) The experts provide following decision

ma-trixes using different linguistic term sets (see

Table 1):

D1¼

a8 b4 b3 c1
a5 b5 b4 c3
a7 b6 b3 c3
a3 b4 b5 c2
2

6
6
6
6
6
4

3
7
7
7
7
7
5
,

D2¼

b5 a2 c1 b2
b3 a4 c3 b3
b5 a6 c2 b3
b4 a3 c3 b2

2

6
6
6
6
6
4

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D3¼

b5 a2 c2 c1
b4 a4 c3 c3
b6 a7 c3 c3
b3 a5 c4 c0
2

6
6
6
6
6
4

3
7
7
7
7
7

5
.

(2) After the experts provide their linguistic
assess-ment, the basic linguistic term set ST is

determined. In this problem, because there is
only one linguistic term set with maximum
granularity, then ST¼{a0,a1,y,a8} ¼ {S0,S1,

y,S₈}. Next, the functions t and w are applied
to transform decision matrices Dk,(k ẳ 1,2,3)
into Dkẳ xkijịnn. The result is given ad follows:

D1ẳ

s8; 0:00ị s4; 0:37ị s4; 0:00Þ ðs2; :0:02

ðs5; 0:00Þ ðs7; 0:39Þ ðs4; 0:37Þ ðs6; 0:02

ðs7; 0:00Þ ðs4; 0:00Þ ðs4; 0:00Þ ðs6; 0:02Þ

ðs3; 0:00Þ ðs4; 0:37Þ s7; 0:39ị s4; 0:02ị

2
6
6
6
6
4

3
7
7
7
7
5,

D2ẳ

s7; 0:39ị s2; 0:00ị s2; 0:02ị s3; 0:39Þ

ðs4; 0:00Þ ðs4; 0:00Þ ðs6; 0:02Þ ðs4; 0:00Þ

ðs7; 0:39Þ ðs6; 0:00Þ ðs4; 0:02Þ ðs4; 0:00Þ

ðs4; 0:37Þ ðs3; 0:00Þ ðs6; 0:02 s3; 0:39ị

2
6
6
6
6
4

3
7
7
7
7

5,

D3ẳ

s7; 0:39ị s2; 0:00ị s4; 0:02ị s2; 0:02ị

s4; 0:37ị ðs4; 0:00Þ ðs6; 0:02Þ ðs6; 0:02Þ

ðs7; 0:50Þ ðs7; 0:00Þ ðs6; 0:02Þ ðs6; 0:02Þ

ðs4; 0:00Þ ðs5; 0:00Þ ðs7; 0:34Þ ðs1; 0:33Þ

2
6
6
6
6
4

3
7
7
7
7
5

(3) Suppose the experts have equal importance.
Integrate Dk,k ¼ 1,2,y,K into objective

deci-sion matrix D ¼ (xij))m  n according to the

similarity degree based aggregation algorithm.

D ẳ

s7; 0:04ị s3; 0:31ị s3; 0:41Þ ðs2; 0:21Þ

ðs4; 0:45Þ ðs5; 0:25Þ ðs6; 0:49Þ ðs5; 0:41Þ

ðs7; 0:04Þ ðs6; 0:28Þ ðs5; 0:39Þ ðs5; 0:41Þ

ðs4; 0:20Þ ðs4; 0:15Þ ðs7; 0:35Þ ðs2; 0:47Þ

2
6
6
6
6
4

3
7
7
7
7
5

(4) By interviewing vendors, examining the
ven-dor’s demonstrations, collecting detailed
infor-mation, the member m1,m2,m3 in project team

finally gives the partial order relation of
candidate ERP systems respectively based on
his own subjective judgment and knowledge:

Y1_{ẳ f3; 1ị; 3; 4ị; f2; 4ịg;} _Y2_{ẳ f3; 4ị; 3; 2ị; 1; 4ịg,}

Y3ẳ f3; 1Þ; ð2; 4Þ; ð3; 2Þ; ð2; 1Þg.

The set of preference relations on alternatives
provided by all members is

Y ẳ f3; 2ị; ð3; 4Þ; ð2; 4Þ; ð1; 4Þ; ð3; 1Þ; ð2; 1Þg.
The important degree of all alternative pairs
in Y is calculated respectively as follows:
m32 ¼

2

3; m34 ¼
2

3; m24¼
2
3,
m14 ¼

1

3; m31 ¼

2

3; m21¼
1
3.

(5) Let h ¼ 1; solving following linear programming
problem:

min B ¼ l32ỵl34ỵl24ỵl14ỵl31ỵl21
s:t: 48:62o139:43o2ỵ28:83o354:87o4

ỵ9:07v1ỵ8:63v24:6v3ỵ14:07v4X1
19:84o16:77o2ỵ6:07o3ỵ3:45v1
ỵ1:29v2ỵ1:20v3l32p0

23:41o110:33o2ỵ15:31o315:44o4
ỵ4:32v1ỵ2:09v22:72v3ỵ3:92v4l34p0

3:58o13:56o2ỵ9:24o315:44o4
ỵ0:87v1ỵ0:80v21:54v3ỵ3:92v4l24p0

11:71o1ỵ3:33o2ỵ10:86o3ỵ0:41o4
ỵ2:16v10:97v22:16v30:16v4l14p0

9:92o15:11o26:24o38:13o41:93v1

ỵ1:37v2ỵ1:40v3ỵ2:12v4l21p0
6:99o26:42o316:26o4ỵ4:04v2
ỵ1:60v3ỵ4:27v4l31p0

o1X0; o2X0; o3X0; o4X0

Table 1

Linguistic term sets and associated semantics of labels
Label set A Label set B Label set C
a0¼(0,0,0.12) b0¼(0,0,0.16) c0¼(0,0,0.25)

a1¼(0,0.12,0.25) b1¼(0,0.16,0.33) c1¼(0,0.25,0.5)

a2¼(0,12,0.25,0.37) b2¼(0.16,0.33,0.5) c2¼(0.25,0.5,0.75)

a3¼(0.25,0.37,0.5) b3¼(0.33,0.5,0.66) c3¼(0.5,0.75,1)

a4¼(0.37,0.5,0.62) b4¼(0.5,0.66,0.83) c4ẳ(0.75,1,1)

a5ẳ(0.5,0.62,0.75) b5ẳ(0.66,0.83,1)

a6ẳ(0.62,0.75,0.87) b6ẳ(0.83,1,1)

a7ẳ(0.75,0.87,1)

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o1ỵo2ỵo3ỵo4ẳ1

0pv1p8o1; 0pv2p8o2; 0pv3p8o3,
0pv4p8o4

l32X0; l₃₄X0; l₂₄X0; l₁₄X0; l₃₁X0; l₂₁X0
We can obtain optimal value 0.8323 and

optimal solution

o

1¼0:0807; o


2¼0:8576; o


3¼0:0376; o


4¼0:0242,

v₁¼0:6452; v₂¼3:8384; v₃¼0; v₄¼0.

From Eq. (11), we can obtain the positive
ideal solution:

x_{ẳ fs}

8; 0:00ị; s5; 0:44ị; s0; 0:00ị; ðs0; 0:00Þg

(6) The distance between each alternative and
positive ideal solution is calculated as follows:
d1¼3.3776, d2¼2.9294, d3¼2.9004, d4¼

3.3273, Based on this distance, the ranking
order of alternative isx3x2 x4x1. So x3is

the most suitable ERP systems for the
organiza-tion.

5. Conclusions

Selecting a suitable ERP system is the basis of
implementing ERP project successfully. This paper
presents a new model for ERP selection, which is
based on linguistic information processing. The
main characteristics of this model are: (1) the
linguistic 2-tuple representation model and
comput-ing model are taken for dealcomput-ing with multi-granular
linguistic assessment information. It overcomes the
drawback of the loss of information in the classical
linguistic computational models such as the
seman-tic model and the symbolic model, (2) In the study, a
similarity degree based algorithm is proposed to
aggregate the information about ERP systems from
external professional organizations. It make
aggre-gation results reflect the collective opinions more
reasonably and more objective, (3) The weights of
attributes are determined by solving linear
pro-gramming. They need not be provided by project
team in advance, (4) It combines the objective
information obtained from external professional
organization and the subjective information from
the member of project team. Therefore, the decision
result is more credible. In a word, the method
proposed in this paper provides an effective tool for

dealing with ERP system selection.

Acknowledgements

This work was supported by NSFC (70121001,
70571063, 70472739). We are grateful to the
anonymous referees for their valuable comments.

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