 Open Access
 Total Downloads : 26
 Authors : M. Shanmuganathan , K. Kajendran ,
 Paper ID : IJERTCON091
 Volume & Issue : PECTEAM – 2018 (Volume 6 – Issue 02)
 DOI : http://dx.doi.org/10.17577/IJERTCON091
 Published (First Online): 17042018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Flaws of Quantification Method as applied to Software Requirements Prioritization
M. Shanmuganathan 1 and K. Kajendran 2
1 Assistant Prof(GI), Dept of C.S.E, Panimalar Engineering College,
2 Associate Prof., Dept of C.S.E, Panimalar Engineering College,
Abstract – This paper deals with decisionmaking using the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) , one of the Multi Criteria Decision Making (MCDM) methods, which was originally developed by Hwang and Yoon in 1981with further developments by Yoon in 1987 and Hwang, Lai and Liu in 1993. It is a goal based approach for finding the alternative that is closest to the optimal solution . In this method, alternatives are graded based on optimal solution or alternative similarity. Optimal solution is a solution that is the best or perfect from any aspect that does not exist practically and tries to approximate it. Basically, for measuring similarity of an alternative to optimal level and nonoptimal, we consider distance of the alternative from optimal and nonoptimal solution. It explains the usefulness of TOPSIS in decision making, quantification of data, solving complex problems, besides touching upon some basic concepts, ideas, benefits, and drawbacks of TOPSIS. The paper includes : I.Introduction, II. Algorithm of TOPSIS, III. Numerical Example, IV. Phenomenon of Rank Reversal, V. Conclusion, and VI. References .
Keywords : MultiCriteria DecisionMaking, Technique for Order Preference by Similarity to Ideal Solution , Decision Making, Rank Reversal.

INTRODUCTION :
Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is an easily understandable, and a systematic MultiCriteria DecisionMaking (MCDM) technique , which was introduced by Hwang and Yoon in the year 1981[1][2][4][8][9][10] with further developments by Yoon in 1987 and Hwang[11][12], Lai and Liu in
1993[5]. This technique is based on the concept that the chosen alternative should have the shortest distance from the positive optimal solution and the longest distance from the negative optimal solution. If an alternative is more similar to optimal solution , it has a higher grade[11] This principle has been also suggested by Zeleny (1982) and Hall(1989) and it has been enriched by Yoon(1987) and Hwang,Lai and Liu (1993).It defines m x n matrix, m alternatives and n criteria and assigns priority to alternatives. It is purely a goal based approach for finding the alternative that is closest to the ideal solution . It is simple to use and takes into account all types of criteria (subjective and objective). It reduces a huge complex problem into a more structured format and facilitates a more practical approach. The computation processes are straightforward. It is applied in many Engineering , Scientific, and other commercial fields[6]. A Decision Maker, who can understand the entire domain of the problem and who has the knowledge of the domain, can use this method without any difficulty. Thus TOPSIS can be considered to be one of the MultiCriteria Decision Making Methods for solving certain complex problems.

ALGORITHM OF TOPSIS:
The basic TOPSIS technique consists of the following steps :
Step (1) : Construct m x n matrix for alternative performance with respect to criteria available, m denotes the number of alternatives and n denotes number of criteria. The structure of the matrix can be expressed as
C1 C2 . . Cn
A1 x11 x12 . . x1n
D = A2 x21 x22 . . x2n
. . . .
Am xm1 xm2 . . xmn
Where Ai denotes the possible alternatives i = 1 .. m , Cj denotes the possible criteria relating to alternative performance j=1..n, Xij is an exact value indicating the performance rating of each alternative Ai with respect to each criterion Cj
Step (2) : Calculate the normalized m x n matrix R(=rij). The normalized value rij is calculated as
rij =xij / ( xij2 ) for i=1 ..m, j=1 .. n
Step(3) : Calculate the weighted normalized matrix by multiplying the normalized decision matrix by its associated weights. The weighted normalized value Vij is calculated as Vij = Wjrij, for i=1 ..m, j=1 .. n where wj represents the weight of the jth criterion
1
Step(4) : Determine the ideal and negative ideal solutions V+= {V +,,Vn+}={(max Vij  j J), (min Vij  j J1)}
1 ij ij
V = {V ,.,Vn}={(min V  j J), (max V  j J1)} Where J is associated with benefit criteria and J1 is associated with cost criteria
i i
Step(5) : Calculate the separation measures, using the m dimensional shortest distance. The separation of each alternative from the ideal solution (D +) is given as D + =(
j
i
( Vij – V + )2 , i=1..m, j=1..n. Similarly, the separation of each alternative from the negative ideal solution (D ) is as follows
i
j
D – =( ( Vij – V – )2 , i=1..m, j=1..n
i
Step(6) : Determine the relative closeness to the ideal solution and rank the preferences. The relative closeness of the alternative Ai with respect to V+ can be expressed as Ci=Di/(Di+ + D ), i=1.m , where Ci index value lies between 0 and 1. The higher the index value, the better the performance of the alternatives will be.
Step(7) : Rank the preference Order

NUMERICAL EXAMPLE
This paper demonstrates that the decisionmaker(software developer) wants to choose a sequence from a set of feasible requirements – Requirement 1(R1), Requirement 2(R2), Requirement 3(R3), Requirement 4(R4) against criteria Criterion1(C1), Criterion2(C2) and Criterion3(C3).
Start of TOPSIS method :
Step (1) : Construct m x n matrix for alternative performance with respect to criteria available, m denotes the number of alternatives (R1, R2, R3, R4) and n denotes number of criteria (C1,C2,C3). In this step the decision makers use the linguistic weighting variables to assess the importance of the criteria. They use the linguistic rating variables to evaluate the rating of alternatives with respect to each criterion. The human feelings are converted into numbers inorder to construct a matrix. The linguistic variables are converted into numerical values by using a 10 point scale
VL 
Very Low 
0 
VP 
Very Poor 
0 
VL 
Very Slow 
0 
L 
Low 
1 
P 
Poor 
1 
L 
Slow 
1 
ML 
Medium Low 
3 
MP 
Medium Poor 
3 
ML 
Medium Slow 
3 
M 
Medium 
5 
F 
Fair 
5 
M 
Fair 
5 
MH 
Medium High 
7 
MG 
Medium Good 
7 
MH 
Medium Fast 
7 
H 
High 
9 
G 
Good 
9 
H 
Fast 
9 
VH 
Very High 
10 
VG 
Very Good 
10 
VH 
Very Fast 
10 
The following table gives a list of alternatives and their respetive criteria. Table 1 shows various alternatives and their respective criteria
The structure of the matrix can be expressed as
Table 1
Alternatives 
C1 
C2 
C3 
R1 
7 
9 
8 
R2 
8 
7 
8 
R3 
9 
6 
8 
R4 
6 
7 
6 
Step (2) : To normalize m x n matrix R(=rij). The normalized value rij is calculated as rij =xij / ( xij2 ) for i=1 ..m, j=1 .. n , It is shown in Table 2
Table 2
Alternatives 
C1 
C2 
C3 
R1 
0.462 
0.614 
0.530 
R2 
0.527 
0.477 
0.530 
R3 
0.593 
0.409 
0.530 
R4 
0.396 
0.477 
0.397 
Step (3): Calculate the weighted normalized matrix by multiplying the normalized decision matrix by its associated weights. The weights of the criteria are assigned as 40% for C1, 30 % for C2, and 30 % for C3. This is based on decisionmakers expertise as indicated in Table 3. The weighted normalized value Vij is calculated as Vij = Wjrij, for i=1 ..m, j=1 .. n as shown in Table 4 where wj represents the weight of the jth criterion
Table 3
Weight( Wj ) 
0.4 
0.3 
0.3 
Alternatives 
C1 
C2 
C3 
R1 
0.462 
0.614 
0.530 
R2 
0.527 
0.477 
0.530 
R3 
0.593 
0.409 
0.530 
R4 
0.396 
0.477 
0.397 
Table 4
Alternatives 
C1 
C2 
C3 
R1 
0.185 
0.184 
0.159 
R2 
0.211 
0.143 
0.159 
R3 
0.237 
0.123 
0.159 
R4 
0.158 
0.143 
0.119 
Step (4) : Determine the positive and negative ideal solutions . For positive ideal solution is shown in Table 5
Table 5 V+ = {0.237,0.184,0.119}
Alternatives 
C1 
C2 
C3 
R1 
0.185 
0.184 
0.159 
R2 
0.211 
0.143 
0.159 
R3 
0.237 
0.123 
0.159 
R4 
0.158 
0.143 
0.119 
For negative ideal solution is shown in Table 6
Table 6 V1 = {0.160,0.123,0.159}
Alternatives 
C1 
C2 
C3 
R1 
0.185 
0.184 
0.159 
R2 
0.211 
0.143 
0.159 
R3 
0.237 
0.123 
0.159 
R4 
0.158 
0.143 
0.119 
Step(5) : Calculate the separation measures, using the m dimensional shortest distance. The separation of each alternative from the positive ideal solution (Di+) is shown in Table 7 , Di+ =( ( Vij Vj+ )2 , i=1..m, j=1..n
Table 7
Alternatives 
C1 
C2 
C3 
( ( Vij Vj + )2 
R1 
(0.185 – 0.237)2 
(0.184 – 0.184)2 
(0.159 – 0.119)2 
0.066 
R2 
(0.211 0.237)2 
(0.143 – 0.184)2 
(0.159 – 0.119)2 
0.063 
R3 
(0.237 – 0.237)2 
(0.123 – 0.184)2 
(0.159 – 0.119)2 
0.073 
R4 
(0.158 – 0.237)2 
(0.143 – 0.184)2 
(0.119 – 0.119)2 
0.089 
i
i ij j
Similarly, the separation of each alternative from the negative ideal solution (D ) is shown in Table 8, D – =( ( V – V – )2 , i=1..m,j=1..n
Table 8
Alternatives 
C1 
C2 
C3 
( ( Vij Vj – )2 
R1 
(0.185 – 0.158)2 
(0.184 – 0.123)2 
(0.159 – 0.159)2 
0.067 
R2 
(0.211 0.158)2 
(0.143 – 0.123)2 
(0.159 – 0.159)2 
0.057 
R3 
(0.237 – 0.158)2 
(0.123 – 0.123)2 
(0.159 – 0.159)2 
0.079 
R4 
(0.158 – 0.158)2 
(0.143 – 0.123)2 
(0.119 – 0.159)2 
0.045 
i i
Step(6) : Determine the relative closeness to the ideal solution . The relative closeness of the alternative Ai with respect to V+ can be expressed as Ci=D /(Di+ + D ), i = 1.m where Ci index value lies between 0 and 1. The higher the index value, the better the performance of the alternatives will be.
Table 9
Alternatives 
+ + D ) Ci=Di/(Di i 
R1 
0.067/(0.066+0.067) = 0.504 
R2 
0.057/(0.063+0.057) = 0.475 
R3 
0.079/(0.073+0.079) = 0.520 
R4 
0.045/(0.089+0.045) = 0.336 
Step (7) : Rank the preference Order
Overall relative closeness and Rank of alternatives is shown in Table 10
Table 10
Alternatives 
Result 
Rank 
R1 
0.504 
2 
R2 
0.475 
3 
R3 
0.520 
1 
R4 
0.336 
4 
R3>R1>R2>R4
(IV)Phenomenon of Rank Reversal :
As already mentioned TOPSIS suffers from the drawback of rank reversal. If a new alternative (new requirement) R5 is added, then the following will be the judgement matrix, with four alternatives in terms of criterion: If the new alternative R5 is added which is similar to R3, then The following will be the judgement matrix, with five alternatives in terms of criterion shown in Table 11
Step (1) :
Table 11
Alternatives 
C1 
C2 
C3 
R1 
7 
9 
8 
R2 
8 
7 
8 
R3 
9 
6 
8 
R4 
6 
7 
6 
R5 
9 
6 
8 
Step (2): To normalize m x n matrix R(=rij). The normalized value rij is calculated as rij =xij / ( xij2 ) for i=1 ..m, j=1 .. n , It is shown in Table 12
Table 12
Alternatives 
C1 
C2 
C3 
R1 
0.397 
0.568 
0.468 
R2 
0.454 
0.442 
0.468 
R3 
0.510 
0.379 
0.468 
R4 
0.340 
0.442 
0.351 
R5 
0.510 
0.379 
0.468 
Step (3): Calculate the weighted normalized matrix by multiplying the normalized decision matrix by its associated weights. The weights of the criteria are assigned as 40% for C1, 30 % for C2, 30 % for C3. This is based on decisionmakers expertise as indicated in Table 3. The weighted normalized value Vij is calculated as Vij = Wjrij, for i=1 ..m, j=1 .. n as shown in Table 13 where wj represents the weight of the jth criterion
Table 13
Weight(Wj) 
0.4 
0.3 
0.3 
Alternatives 
C1 
C2 
C3 
R1 
0.397 
0.568 
0.468 
R2 
0.454 
0.442 
0.468 
R3 
0.510 
0.379 
0.468 
R4 
0.340 
0.442 
0.351 
R5 
0.510 
0.379 
0.468 
Table 14
Alternatives 
C1 
C2 
C3 
R1 
0.159 
0.170 
0.140 
R2 
0.182 
0.133 
0.140 
R3 
0.204 
0.114 
0.140 
R4 
0.136 
0.133 
0.105 
R5 
0.204 
0.114 
0.140 
Step (4) : Determine the positive and negative ideal solutions For positive ideal solution is shown in Table 15
Table 15 V+ = {0.204,0.170,0.105}
Alternatives 
C1 
C2 
C3 
R1 
0.159 
0.170 
0.140 
R2 
0.182 
0.133 
0.140 
R3 
0.204 
0.114 
0.140 
R4 
0.136 
0.133 
0.105 
R5 
0.204 
0.114 
0.140 
For negative ideal solution is shown in Table 16
Table 16 V1 = {0.136,0.114,0.140}
Alternatives 
C1 
C2 
C3 
R1 
0.159 
0.170 
0.140 
R2 
0.182 
0.133 
0.140 
R3 
0.204 
0.114 
0.140 
R4 
0.136 
0.133 
0.105 
R5 
0.204 
0.114 
0.140 
i i ij j
Step(5) : Calculate the separation measures, using the m dimensional shortest distance. The separation of each alternative from the positive ideal solution (D +) is shown in Table 17 , D + =( ( V – V + )2 , i=1..m, j=1..n
Table 17
Alternatives 
C1 
C2 
C3 
( ( Vij Vj + )2 
R1 
(0.159 – 0.204)2 
(0.170 – 0.170)2 
(0.140 – 0.105)2 
0.057 
R2 
(0.182 – 0.204)2 
(0.133 – 0.170)2 
(0.140 – 0.105)2 
0.055 
R3 
(0.204 – 0.204)2 
(0.114 – 0.170)2 
(0.140 – 0.105)2 
0.066 
R4 
(0.136 – 0.204)2 
(0.133 – 0.170)2 
(0.105 – 0.105)2 
0.077 
R5 
(0.204 – 0.204)2 
(0.114 – 0.170)2 
(0.140 – 0.105)2 
0.066 
i
i
j
Similarly, the separation of each alternative from the negative ideal solution (D ) is shown in Table 18, D – =( ( Vij V – )2 , i=1..m, j=1..n
Table 18
Alternatives 
C1 
C2 
C3 
( ( Vij Vj – )2 
R1 
(0.159 – 0.136)2 
(0.170 – 0.114)2 
(0.140 – 0.140)2 
0.061 
R2 
(0.182 – 0.136)2 
(0.133 – 0.114)2 
(0.140 – 0.140)2 
0.050 
R3 
(0.204 – 0.136)2 
(0.114 – 0.114)2 
(0.140 – 0.140)2 
0.068 
R4 
(0.136 – 0.136)2 
(0.133 – 0.114)2 
(0.105 – 0.140)2 
0.040 
R5 
(0.204 – 0.136)2 
(0.114 – 0.114)2 
(0.140 – 0.140)2 
0.068 
Step(6) : Determine the relative closeness to the ideal solution . The relative closeness of the alternative Ai with respect to V+ can be expressed as Ci=D /(Di+ + D ), i=1.m where Ci index value lies between 0 and 1. The higher the index value, the better the
i i
performance of the alternatives will be. It is shown in Table 19.
Table 19
Alternatives 
+ + D ) Ci=Di/(Di i 
R1 
0.061/(0.057+0.061) = 0.517 
R2 
0.050/(0.055+0.050) = 0.476 
R3 
0.068/(0.066+0.068) = 0.507 
R4 
0.040/(0.077+0.040) = 0.342 
R5 
0.068/(0.066+0.068) = 0.507 
Step (7) : Rank the preference Order
Overall relative closeness and Rank of alternatives is shown in Table 20
Table 20
Alternatives 
Result 
Rank 
R1 
0.517 
1 
R2 
0.476 
3 
R3 
0.507 
2 
R4 
0.342 
4 
R5 
0.507 
2 
R1 > R3= R5 > R2 > R4
V. CONCLUSION :
TOPSIS is one of the MultiCriteria DecisionMaking methods (MCDM) and it has been applied in different fields despite certain drawbacks . In this numerical example discussed above when four alternatives are considered, the rank becomes R3 > R1 > R2 > R4 for the first empirical analysis. When a new alternative R5 is added to an existing alternative which is similar to R3, then, the rank becomes R1 > R3 = R5 > R2 > R4. This paper clearly indicates that rank reversal exists when new alternatives are added to or deleted from an existing alternatives. According to literature available on TOPSIS, the lnguistic variables are converted into numerical values. In other words, Human feelings are converted into numbers, i.e., quantified in order to suit this scale[7]. Human feelings differ from person to person. Psychologically, human feelings cannot quantify [3]. Despite certain drawbacks, this method cannot be ignored because this technique provides an easy, understandable, proper, straight forward computation besides being a
systematic and meaningful method for academic community to make better decisions.
VI . REFERENCES :

Alecoskelemenis, Dimitrios Askounis (2009). A new Topsis based multicriteria approach to personnel selection, Expert selection, Expert Systems with Applications, vol.9, no.2, pp.641646.

Dipendra Nath Ghosh.(2011). Analytic Hierarchy Process and Topsis Method to Evaluate Faculty performance in engineering Education , UNIASCIT, vol.1(2), pp.6370.

Hand, D.J.(1996). Statistics and the Theory of Measurement, Journal of Royal Statistical Society. Series A(Statistics in Society), vol.159, no.3, pp. 445492.

Hwang,C.L., Yoon,K (1981). Multiple Attribute Decision Making : Methods and Applications. Newyork :Springer Verlag.

Hwang., C.L., Lai, Y.J; Liu, T.Y.(1993). A new approach for multiple objective decision making, Computers and Operatioanl Research, vol. 20, Issue 8, pp. 889899.

Majid Behzadian, Khanmohammadi,S, Otaghsara, Morteza Yazdani, Joshua Ignatius.(2012). A stateof the art survey of Topsis applications, Expert systems with Applications, An
International Journal archive,vol. 39, Issue 17, pp. 12727
13074.

Pelin Alcan, Huseyin Bashgil.(2011). A Facility Location Selection Problem by Fuzzy Topsis, 15th International Research/Expert Conference Trends in Development of Machinery and Associated Technology,TMT,Prague,Czech Republic.

Pema Wang chen Bhutia, Ruben Phipon.(2012) . Application of AHP and TOPSIS method for supplier selection problem, IOSR Journal of Engineering ,vol. 2, Issue 10, pp.4350.

Serkan Balli, Serdar Korukoglu .(2009). Operating System Selection using Fuzzy AHP and TOPSIS Methods, Mathematical and Computational Applications, vol.14, no.2, pp.119130.

Shahroudi,K and Rouyddel,H..(2012). Using a multicriteria decisionmaking approach (ANPTOPSIS) to evaluate suppliers in Irans auto industry, International Journal of Applied Operational Research, vol.2, no.2, pp. 3748.

Yoon,K.(1987). A Reconciliation among discrete compromise situations., Journal of Operational Research Society, vol 38, no.3, pp.277 286.

Zoran Markovic,(2010). Modification of Topsis Method for solving of Multicriteria Tasks,Yugoslav Journal of Operations Research,vol.20, no.1, pp.117143.