ERP Software Selection Using The Rough Set And TPOSIS Methods

ERP Software Selection Using The Rough Set And TPOSIS Methods

Under Fuzzy Environment

Huang Huiqun, and Sun Guang

Information Management Department, Hunan University of Finance and Economics, No.

139, Fenglin 2nd Road, Changsha, 410205, China

[email protected]

Abstract

Improperly selected ERP software may have an impact on the time required, and the costs and market share of a company, selecting the best desirable ERP software has been the most critical problem for a long time. On the other hand, selecting ERP software is a multiple-criteria decision-making (MCDM) problem, and in the literature, many methods have been introduced to evaluate this kind of problem, which has been widely used in MCDM selection problems. In this paper, an integrated approach of ERP software selection analytic hierarchy process improved by rough sets theory (Rough-AHP) and fuzzy TOPSIS method is proposed to obtain final ranking.

Keywords:

Enterprise Resource Planning ; Software Selection ; Rough Set ; TOPSIS

1. Introduction

Any ERP software in market cannot fully meet the needs and expectations of companies, because every company runs its business with different strategies and goals. ERP vendors use different hardware platforms, databases, and operation systems, and some ERP software is only compatible with some companies’ databases and operation systems. Therefore, one of the most critical issues in implementation of an ERP system is the selection of the appropriate software to be used. The importance of the actual software selection Influential factors must not be underestimated. In our study, These identified five important dimensions of performance are Risk, Quality, Effectiveness, Efficiency, and customers satisfaction degree.

2. Analytic hierarchy process improved by rough set theory (Rough-AHP)

AHP, developed by Saatty (1980), addresses how to determine the relative importance of a set activities in different decision-making processes. The AHP method is based on three principles: structure of the hierarchy, the matrix of pairwise comparison ratios and the method for calculating weights. But, AHP is strongly connected to human judgment and pairwise comparisons in AHP may cause evaluator’s assessment bias which situation makes the comparison judgment matrix inconsistent. In this paper the concept of attribute significance in rough sets theory is utilized to solve evaluation bias problem in AHP. Conditional entropy and attribute significance concepts in rough sets theory are utilized in AHP to improve the judgment consistency.

Formally, a data table is the 4-tuple S=(U,R,V,f) where U is a finite set of objects (universe);R = C



D is a set of attributes, subsets C and D are the condition attribute set and the decision attribute set,respectively; Vr is domain of the attribute r, V=



Vr



A and f:U



A



V is a total function such that f(x, r)



Vr for each r



R, x



U, called information function.

Definition 1.Entropy H(P) of knowledge P (attributes set) is defined as

1

( )

n

( ) log ( )

_{i i} i

H p

p X

p X



 



(1)

Where

p X

(

i

)



XiU and p(Xi)donates the probability of Xi when p is on the partition

1 2

{ ,

, ,

_n

}

X



X X

_

X

_{of universe U, i=1,2,…,n.}

Definition 2. conditional entropy

H Q P

(

)

which knowledge

Q U IND Q

(

( )) { , , , }



Y Y

1 2



Y

m _is relative to knowledge

Q U IND P

(

( )) { ,



X X

1 2

, ,



X

n

}

_{is defined as}

1 1

(

)

n

( )

i m

(

j i

) log (

j i

)

i j

H Q P

p X

p Y X

p Y X

 

 





(2)

Where

p Y X

(

j i

)

_{is conditional probability, i=1,2,…,n,j=1,2,…,m.}

Definition 3.Suppose that decision table S=(U,R, V, f), R =

C D



, subsets C and D are the condition attribute set and the decision attribute set, respectively, attribute subset A



C. The attribute significance Sig(a, A, D) of attribute a



C/A is defined as

Sig(a, A, D)=

H D A

(

)



H D A

(



{ })

a

(3) Given attribute subset A,the greater the value of Sig(a, A, D),the more important attribute a is for decision D.

3. Fuzzy TOPSIS

TOPSIS method is a technique for order performance by similarity to ideal solution from a finite set of points. The main rule is that the best alternative would be the “shortest” distance from the positive ideal solution and the “furthest” distance from the negative ideal solution. But in real life, measurement by using crisp values is not always possible. For this reason, the fuzzy TOPSIS method is very suitable for solving real life application problems under a fuzzy environment.

Definition 4.Let ~ 1 2 3

( , , )

a



a a a

_and ~ 1 2 3

( , , )

b



b b b

_{be two triangular fuzzy numbers then the vertex} method is defined to calculate the distance between them

~ ~ 2 2 2 1 3 1 1 2 2 3 3

( , )

[(

)

(

)

(

) ]

d a b



a



b



a



b



a



b

(4) Definition 5. Considering the different importance values of each criterion, the weighted normalized fuzzy-decision matrix is constructed as.

~

[ ]

ij n j

V



v



_ ,i=1,2,…,n,j=1,2,…,J, (5) Where

( )

ij ij i

v

_



X



W

A set of performance ratings of Aj,(j=1,2,…,J)with respect to criteria Ci,(i=1,2,…,n)called

X





( ,

x i



ij



1, 2, , ,



n j



1, 2, , )



J

A set of importance weights of each criterion Wi(i=1,2,…,n).

Fuzzy TOPSIS steps can be outlined as follows:

Step1:choose the linguistic rating

( ,

x i



ij



1, 2, , ,



n j



1, 2, , )



J

_{for alteratives with respect to} criteria. The fuzzy linguistic ratin(

~

ij

x

_{) preserves the property that the ranges of normalized triangular} fuzzy numbers belong to [0,1];thus, thers is no need for normalization.

Step 2:calculate the weighted normalized fuzzy decision matrix. the weighted normalized value

~

ij

v

caluated by eq.(5).

Step 3:inentify positive-ideal (A*) and negative ideal (A-) solutions. The fuzzy positive-ideal

solution(FPIS, A*)and the fuzzy negative-ideal solution(FNIS, A-)are shown in the following equations:





* * * * 1 2,

{ , ..., }

_i

max

_ij

min

_ij j j

A

v v

v







_



v i

I



_



v i

I















_{ }

_









(6) i=1,2,…,n,j=1,2,…,J, 1 2,













{ , ..., }

_i

min

_ij

max

_ij j j

A





v v

_{ }

 

v

_





v i



I





v i



I



(7) i=1,2,…,n,j=1,2,…,J,

where

I



is associated with benefit criteria and

I



is associate with cost criteria.

Step 4:caluate the distance of each alternative from

A

* and

A

 using the following equations:

* * 1

( , )

n j ij i j

D

d v v







 

j=1,2,…,J (8) 1

( , )

n j ij i j

D



d v v

 





_{ }

j=1,2,…,J (9) Step 5:caluate similarities to ideal solution

* j j j j

D

CC

D

D

 





(10)

Step 6:rank preference order choose an alternative with maximum

*

j

CC

or rank alternatives according to

*

j

CC

in descending order.

4. A new methodology for ERP software selection

4.1. Define the criteria for ERP system selection

If there are more ERP alternatives in the list than expected, a pre-selection process should be used to reduce the number of alternatives to an acceptable level (three or four) so that the selection process will

not be too lengthy. We selected four ERP system selection firms(such as CA-MANMAN/X, BAAN, SSA-BPCS and SAP R/3),to evaluate their selection indicators and their weights in total score, so we interviewed with ERP users, the firms’s and corporation’s senior management cadre. Expectations from a ERP software selection scale include particularly low risk, high quality, flawless product, reliability, delivery on time. To develop fuzzy environment selecton model, we first identified various Influential factors of ERP system selection and the corresponding indicators, that are used to evaluate those firms under aforementioned expectations. These identified five important Influential factors of software selection are Risk, Quality, Effectiveness, Efficiency, and users satisfaction degree. We arranged the hierarchy structure of the selection indicators in Fig.1.

4.2. Calculate the weights of criteria

After forming the hierarchy of the problem, decision table is built. In decision table (i.e. Table 1) rows indicate the distinct objects, and columns indicate the different attributes(i.e. selection indicators) considered. Initially decision column is empty. The risk, quality, effectiveness, efficiency and users satisfaction degree criteria are rated using the 1, 2, 3 values. Only for the Risk criteria 1 means low, 2 means medium and 3 means high whereas these mean the contrary for other criteria; 1 high, 2 medium and 3 low. Secondly, we can make a table that lists different combinations of criteria rates before evaluation process. In Table 1, we list 24 different combinations. Then the table is given to evaluation team to make a decision. The number “1” in decision column represents “selection approves” and the number “0” represents “selection disapproves”.

Figure 1. The hierarchy of the study.

Selection of the best ERP firms

Risk Quality Effectiveness Efficiency

Users satisfaction

Table 1. Decision table about risk, quality, effectiveness, efficiency and occupational satisfaction U Risk (a) Quality (b) Effectiveness (c) Efficiency (d) Occupational Satisfaction(e) Decision (D) 1 2 3 1 3 2 0 2 1 2 2 3 3 1 3 3 1 1 2 2 0 4 1 2 1 1 3 1 5 2 3 3 2 2 0 6 2 3 2 2 3 0 7 1 3 3 3 2 0 8 1 2 2 2 2 1 9 2 1 2 3 3 1 10 3 2 1 1 3 0 11 1 2 2 1 3 1 12 3 2 2 2 2 0 13 2 2 3 3 3 0 14 1 3 1 3 2 1 15 2 2 2 2 2 1 16 2 2 3 2 3 0 17 2 2 2 2 3 1 18 2 2 2 3 3 0 19 2 1 3 2 2 1 20 2 1 1 2 2 1 21 2 2 2 1 3 1 22 2 3 1 2 2 0 23 2 2 3 2 1 1 24 2 1 3 3 3 0

For the decision table of table 1,we can get criteria significances of risk, quality, effectiveness, efficiency, and occupational satisfaction by the following process:

U\IND{a,b,c,d,e}={{1},{2},{3},{4},{5},…, {20},{21},{22},{23},{23}}, U\IND{D}={{2,4,8,9,11,14,15,17,19,20,21,23},{1,3,5,6,7,10,12,13,16,18,22,24}}={D1,D2}, U\IND{b,c,d,e}={{4,10},{8,12,15},{3,20},{2,18},{1,14}} ={X1,X2,X3,X4,X5}, P(X1)=2/24,P(D1\ X1)=1/2, P(D2\ X1)=1/2; P(X2)=3/24,P(D1\ X2)=1/3, P(D2\ X2)=1/2; P(X3)=2/24, P(D1\ X3)=1/2, P(D2\ X3)=1/2; P(X4)=2/24, P(D1\ X4)=1/2, P(D2\ X4)=1/2; P(X5)=2/24, P(D1\ X5)=1/2, P(D2\ X5)=1/2; Sig(a,{b,c,d,e},{D})=H({D}\{b,c,d,e})-H({D}\{a,b,c,d,e}) = 2 1 1 1 1 3 1 1 2 2

( log log ) 4 ( log log )

24 2 2 2 2 24 4 3 3 3

    

=0.135

We obtain the significance of attribute a (i.e. risk criterion) is 0.135.By the similar process, we also can get the significance of attribute b(i.e. quality criterion)is 0.100 and the significance of attribute c(i.e. effectiveness criterion) is 0.075 and the significance of attribute d(i.e. efficiency criterion) is 0.035 and the significance of attribute e (i.e.users satisfaction degree) is 0.025,respectively.

For risk, quality, effectiveness, efficiency and users satisfaction degree critera, the judgement matrix J is constructed according criteria significance as follows:

1 1.345 1.792 3.899 5.375 0.744 1 1.332 2.899 3.996 0.558 0.751 1 2.176 3 0.256 0.345 0.459 1 1.387 0.186 0.250 0.333 0.725 1 J                 

This matrix is then translated into the largest eigenvalue problem and resulting priority weights of risk, quality, effectiveness, efficiency, occupational satisfaction are found as 0.364,0.271,0.203,0.093 and 0.068,respectively.The largest eigenvalue



max _{is 5.The consistency index(CI) is defined as}

max

1

n

CI

n









_{, (11)}

where n is the rank of judgment matrix. According to the formula of CI, we know that CI=0 for matrix J. This result shows that pairwise comparison matrix constructed by rough sets method possesses complete consistency.

4.3. Evaluate the alternatives with fuzzy TOPSIS and determinate the final rank

During the decision procedure, the evaluation users were asked to establish the decision matrix by comparing alternatives under each of the criteria one by one. Fuzzy Evaluation Matrix formed by the evaluation of alternative is shown in linguistic variables in Table 2.The fuzzy evaluation matrix constructed by linguistic variables is converted to triangular fuzzy numbers, which are equivalent to linguistic variables, as seen in Table 3.After the determination of fuzzy evaluation matrix the next action is to obtain a fuzzy weighted decision table. By using the criteria weights obtained from rough-AHP in this level, the Weighted Evaluation Matrix is established with Eq. (5). The consequent fuzzy weighted decision matrix is presented in Table 4.

Table 2. Linguistic values and fuzzy numbers. Linguistic values Fuzzy numbers

Very low(VL) (0,0,0.2) Low (L) (0,0.2,0.4) Medium (M) (0.2,0.4,0.6) High (H) (0.4,0.6,0.8) Very high (VH) (0.6,0.8,1) Excellent (E) (0.8,1,1)

In relation to (Table 4) the elements

v



ij _{are normalized positive triangular fuzzy numbers and} their ranges are associated with the closed interval [0,1]. Thus, we can determine the fuzzy positive ideal solution(FPIS,

A

* ) and the fuzzy negative-ideal solution (FNIS,A-) as

*

_(1,1,1)

i



_and

(0, 0, 0)

i



_

for benefit criterion, and *

(0, 0, 0)

i



_and i

(1,1,1)



_

for cost criterion. All along the study, risk criteria is defined as cost criteria, whereas quality, effectiveness, efficiency and users satisfaction degree are defined as benefit criteria. For the third phase, the distance of each alternative from

D

* _and

D

 can be calculated by using Eqs. (8) and (9).The last phase elucidates the similarities to an ideal solution by Eq.(10).In order to illustrate steps 3 and 4 calculation,

CC

j calculation is used as an example as follows:

1 * 1 1

1.163

0.232

3.860 1.163

j

CC

D

D

D

 





＝



＝

Similar calculations are done for the other alternatives and the results of fuzzy TOPSIS analyses are summarized in Table 5. Based on

CC

j _{values, the ranking of the alternatives in descending order are} ERP system selection firm 4,2,1,3. These results indicate that ERP system selection firm 4 has the best choice.

Table 3. Fuzzy evaluation matrix for the alternative ERP system selection. C1 C2 C3 C4 C5 A1 (0.6,0.8,1) (0.6,0.8,1) (0.4,0.6,0.8) (0.2,0.4,0.6) (0.8,1,1) A2 (0.4,0.6,0.8) (0.6,0.8,1) (0.4,0.6,0.8) (0.8,1,1) (0.2,0.4,0.6) A3 (0.8,1,1) (0.4,0.6,0.8) (0.6,0.8,1) (0,0.2,0.4) (0,0.2,0.4) A4 (0.2,0.4,0.6) (0.4,0.6,0.8) (0.4,0.6,0.8) (0.6,0.8,1) (0.4,0.6,0.8) weight 0.364 0.271 0.203 0.093 0.068

Table 4. Weighted evaluation for the alternative ERP system selection

C1 C2 C3 C4 C5 A1 (0.218,0.291,0.364) (0.163,0.217,0.271) (0.081,0.122,0.162) (0.019,0.037,0.056) (0.054,0.068,0.068) A2 (0.146,0.218,0.291) (0.163,0.217,0.271) (0.081,0.122,0.162) (0.074,0.093,0.093) (0.014,0.027,0.041) A3 (0.291,0.364,0.364) (0.108,0.163,0.217) (0.122,0.162,0.203) (0.000,0.019,0.037) (0.000,0.014,0.027) A4 (0.073,0.146,0.218) (0.108,0.163,0.217) (0.081,0.122,0.162) (0.056,0.074,0.093) (0.027,0.041,0.054) A* □*1=(0,0,0) □*2=(1,1,1) □*3=(1,1,1) □*4=(1,1,1) □*5=(1,1,1) A- □-1=(1,1,1) □-2=(0,0,0) □-3=(0,0,0) □-4=(0,0,0) □-5=(0,0,0)

Table 5. Rough AHP-fuzzy TOPSIS results. Alternatives Dj* Dj- CCj A1 3.680 1.163 0.232 A2 3.776 1.248 0.248 A3 3.968 1.037 0.206 A4 3.760 1.269 0.252

5. Conclusion and suggestions

For evaluating performance of the ERP system selection firm, rough-AHP and fuzzy TOPSIS are applied. Despite the fact that AHP and Fuzzy TOPSIS have been utilized in many places due to their easy-to-apply features and effectiveness in multi-criteria decision making there has not been any study in the literature about the application and theory of combined rough-AHP and fuzzy TOPSIS. However, in rough-AHP, the qualitative judgment can be quantified to make more intuitionistic comparisons and reduce or eliminate assessment bias in pairwise comparison process. The proposed method is important, because it can be implemented to structures and other areas.

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