_Mai Thi Thu Ha - A model for selecting an ERP system based on linguistic information processing

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 [r]

Information Systems 32 (2007) 1005–1017

A model for selecting an ERP system based on linguistic

information processing

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: liaoxiuwu@mail.xjtu.edu.cn (X Liao) ,

liyuan@mail.xjtu.edu.cn (Y Li) , lubingemail@163.com (B Lu)

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 The models in the first category transforms linguistic assessment information into fuzzy num-bers and uses fuzzy arithmetic based on the Extension Principle to make computations over these fuzzy numbers The use of fuzzy arithmetic increases the vagueness of the results The results obtained by the fuzzy arithmetic are fuzzy numbers that usually do not match any linguistic term in the initial term set, so a linguistic approximation process is needed to express the result in the original

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

One possibility of generating the linguistic term set consists in directly supplying the term set by considering all terms distributed on a scale on which

a total order is defined [39] For example, a

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; 0:5Þ DðbÞ ¼ ðsi; aÞ

a ¼ b  i; a 2 ½0:5; 0:5Þ;

(

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 ) (si,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: Negððsi; aiÞÞ ¼Dðg 

ðD1ðsi; aiÞÞÞwhere siAS ¼ {s0,s1,y,sg} (3) Aggregation of 2-tuple

Fig 1 A set of seven terms with their sementics.

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

z1½ðb1; a1Þ; ðb2; a2Þ; ; ðbn; anÞ

1ðbi; aiÞ

n

Xn i¼1

n

Xn i¼1

bi

!! , where bi¼D1(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

z2½ðb1; a1Þ; ðb2; a2Þ; ; ðbn; anÞ

Pn

i¼1D1ðbi; aiÞwi

Pn

i¼1wi

Pn i¼1biwi

Pn i¼1wi

, where bi¼D1(bi,ai) ¼ i+aI

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

set of linguistic 2-tuples and W ¼ fðr1; a0

1Þ;

ðr2; a0

2Þ; ; ðrn; a0

nÞg be their associated linguistic 2-tuple weights, The 2-2-tuple linguistic weighted

average operator z3is

z3½ððb1; a1Þ; ðr1; a0

1ÞÞ; ððb2; a2Þ; ðr2; a0

2ÞÞ; , ððbn; anÞ; ðrn; a0

nÞÞ

Pn

i¼1D1ðri; a0

iÞ D1ðbi; aiÞ

Pn

i¼1D1ðri; a0

iÞ

!

Definition 6 Let a ¼ {(b1,a1),(b2,a2),y,(bn,an)} and

b ¼ {(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:

dða; bÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Xn

j¼1

D1ðrj; jÞ½D1ðbj; ajÞ D1ðcj; bjÞ2

Pn j¼1D1ðrj; jÞ

v

u

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 jD1ðbj; 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; PK

k¼1k¼1 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; sjk1; ;

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 xk

12    xk

1n

xk

21 xk

22    xk

2n

xk m1 xk m2    xk

mn

2 6 6 6

3 7 7

7.

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 S ;

Decision matrix D1

Decision matrix D2

Decision matrix D K

Preference relation Θ 1

Preference relation Θ 2

Preference relation ΘH

e1

e2

e K

Transfer D k (k = 1, 2 , K ),

into 2-tuple linguistic

decision matrix D k

Calculate group decision matrix

D by means of

Similarity degree based aggregation algorithm

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

H

k =1 Θk

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

m1

m2

m H

Fig 2 The process for selecting an ERP system.

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; ; sjkg

jkg and ST ¼ fs0; s1; ; sgg be two

linguistic term sets and gXgjk, then a

multi-granularity transformation function tSjk S T is defined

as

tSjk S T : Sjk !F ðSTÞ

tSjk S Tðsjki Þ ¼ fðsl; aijkl Þjl 2 f0; 1; ; ggg; 8sjki 2Sjk

aijkl ¼max

y minfmsjk

i ðyÞ; mslðyÞg,

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

msjk

iðyÞ and mslðyÞ are the membership functions

associated to the linguistic terms sjki 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 ðSTÞ ! ½0:g,

wðtSjk S Tðsjki ÞÞ ¼wðfðsl; aijkl Þ,

l ¼ 0; 1; ; ggÞ ¼ D

Pg l¼0laijkl

Pg l¼0aijkl

!

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 Dk¼

ðxk

ijÞnn, where xk

ij; i ¼ 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 xk¼ ðsk; akÞ, where

sk2S and ak2 ð0:5; 0:5 D can be written

explicitly as

Dk¼

ðsk

11; ak

11Þ ðsk

12; ak

12Þ    ðsk

1n; ak 1nÞ

ðsk

21; ak

21Þ ðsk

22; ak

22Þ    ðsk

2n; ak 2nÞ

ðsk m1; ak m1Þ ðsk m2; ak m2Þ    ðsk

mn; ak

mnÞ

2 6 6 6

3 7 7

7.

3.2 Similarity degree based objective information aggregation

After decision matrix Dk¼ ðxkijÞmn; k ¼ 1; 2; ;

K is calculated, respectively We give a similarity

k ¼ 1,2,y,K into objective decision matrix D ¼

ðxijÞÞmn: The aggregation process is carried out in following steps:

(1) Calculating the similarity degree simðxkij; xlijÞ of the assessment values of alternative xi

(i ¼ 1,2y,m) with respect to attribute cj (j ¼ 1,2,y,n) between decision makers ek and

el, 1pk, lpK, k6¼l

The value D1ðxk

ijÞ D1ðxl

ijÞ



ij; al

ijÞ





D1ðsk

ij; al

ijÞjcan be used to measure the distance between xk

ij and xl

ij: Thus the similarity degree simðxkij; xlijÞcan be defined as follows[34]:

sim ðxkij; xlijÞ ¼1  D

1ðxk

ijÞ D1ðxl

ijÞ g











,

where g+1 is the granularity of BLTS ST The range of simðxk; xl

ijÞis the closed interval [0, 1] The closer simðxk; xl

ijÞto 1 the more similar xk and xl

ij are; while the closer simðxk; xl

ijÞto 0 the more distant xk

ij and xl

ij 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 ¼ 1,2,yn), where

SMij¼ ½sim ðxk

ij; xl

ijÞKK, and simðxk

ij; xl

ijÞ ¼1; 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 assessment values of alternative x (i ¼ 1,2,y,m)

with respect to attribute cj(j ¼ 1,2,yn), where

SMijðekÞ ¼

PK

l¼1;l aksim ðxk; xlijÞ

RSMijðekÞ ¼ SMijðekÞ

PK l¼1SMijðelÞ

(4) Calculating the importance degree bkijof decision

maker ek (k ¼ 1,2,y,K) in the aggregation of

the assessment values xl

ij; l ¼ 1; 2; ; K; where

bkij¼ kRSMijðekÞ

PK

l¼1½lRSMijðekÞ

(5) At last, calculating the group decision matrix

D ¼ (xij))m  n by using operator z2 which has

been defined in Section 2, where

xij¼z2½ðs1ij; a1ijÞ; ðs2ij; a2ijÞ; ; ðskij; akijÞ

PK

l¼1D1ðsl

ij; al

ijÞblij

Pn i¼1blij

!

l¼1D1ðslij; alijÞblij

¼ ðbij; aijÞ

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

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

(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 Yk 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 define

ðp;qÞ2Y K

pk

Obviously, mpq¼pk, 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

mpq¼#fmkjðp; qÞ 2 Yk; mk2Eg

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

mkAE}, E ¼ k ¼ 1,2,y,H} In this paper, mpq 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 {l0, l1,y,lh} 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, ejA[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

Vi¼dðxi; xÞ

j¼1

D1ðrj; jÞ½D1ðbij; aijÞ D1ðbj; ajÞ2

Pn j¼1

D1ðrj; jÞ

0 B B

1 C C

1=2

ð2Þ From Eq (2), we can easily get that Vi belongs to interval [0,g] Furthermore Vi¼0, if xi¼x Let

oj¼ D1ðrj; jÞ

Pn j¼1D1ðrj; jÞ; bij¼D1ðbij; aijÞ,

bj¼D1ðbj; ajÞ,

then ojX0; Snj¼1oj¼1 and bij; bj2 ½0; g Eq (2)

can be simplified as

Vi¼ Xn

j¼1

ojðbijbjÞ2

!1=2

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

to measure the distance between xi and positive

ideal solution x:

di¼V2i ¼Xn

j¼1

ojðbijbjÞ2; i ¼ 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:

dp¼Xn

j¼1

dq¼Xn

j¼1

8ðp; 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:

ðdqdpÞ

¼

mpqðdpdqÞ dqodp

(

¼maxf0; mpqðdpdqÞg

ð5Þ 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 d is 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

ðp;qÞ2Y

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

ðp;qÞ2Y

where

ðdqdpÞþ

¼

mpqðdqdpÞ; dqXdp

(

¼maxf0; mpqðdqdpÞg

ð8Þ 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:

ðdqdpÞþ ðdqdpÞ¼mpqðdqdpÞ

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:

ðp;qÞ2Y maxf0; mpqðdpdqÞg

ojX0 j ¼ 1; 2; ; n

Xn j¼1

oj ¼1,

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

lpq ¼maxf0; mpqðdpdqÞg

Then, we have

lpqX0; lpqXmpqðdpdqÞ

Thus, mathematic programming problem (9) can be

transformed into

ðp;qÞ2Y

lpq s:t: G  BXh

mpqðdpdqÞ lpqp0; ðp; qÞ 2 Y

ojX0; j ¼ 1; 2; ; n

Xn

j¼1

oj¼1

0pbjpg; j ¼ 1; 2; ; n

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

(j ¼ 1,2,y,n), the linear programming problem

(10) can be rewritten as follows:

ðp;qÞ2Y

lpq

s:t: Xn

j¼1

ðp;qÞ2Y

mpqðb2qjb2pjÞ

2Xn

j¼1

vj

X ðp:qÞ2Y

mpqðbqjbpjÞ

Xh

Xn

j¼1

oj½mpqðb2pjb2qjÞ

2Xn

j¼1

vj½mpqðbpjbqjÞ

lpqp0ðp; qÞ 2 Y

ojX0; j ¼ 1; 2; ; n

Xn

j¼1

oj¼1

0pvjpgoj; j ¼ 1; 2; ; n

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; o

2; :o

n; v

1; v

2; ; v

nÞ

Furthermore, we can get the positive ideal solution

using following equation:

x¼ ððb1; a1Þ; ðb2; a2Þ; ; ðbn; anÞÞ



1

o

1

 

; D n

 2

o 2

 

; ; D n

 3

o 3

 

After the weight vector o ¼ ðo

1; o

2; ; o

nÞ 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:

di¼Xn j¼1

oj½D1ðbij; aijÞ D1ðbj; ajÞ2

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

3 7 7

7,

D2¼

b5 a2 c1 b2

b3 a4 c3 b3

b5 a6 c2 b3

2 6 6 6

3 7 7

7,