Flaws of Quantification Method as applied to Software Requirements Prioritization

Flaws of Quantification Method as applied to Software Requirements Prioritization

M. Shanmuganathan 1 and K. Kajendran 2

1 Assistant Prof(G-I), Dept of C.S.E, Panimalar Engineering College,

2 Associate Prof., Dept of C.S.E, Panimalar Engineering College,

Abstract – This paper deals with decision-making 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 non-optimal, we consider distance of the alternative from optimal and non-optimal 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 : Multi-Criteria Decision-Making, Technique for Order Preference by Similarity to Ideal Solution , Decision- Making, Rank Reversal.

1. INTRODUCTION :

   Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is an easily understandable, and a systematic Multi-Criteria Decision-Making (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 straight-forward. 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 Multi-Criteria Decision- Making Methods for solving certain complex problems.

2. 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

3. NUMERICAL EXAMPLE

This paper demonstrates that the decision-maker(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 Criterion-1(C1), Criterion-2(C2) and Criterion-3(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

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

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

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 decision-makers 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

Table 4

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}

For negative ideal solution is shown in Table 6

Table 6 V1 = {0.160,0.123,0.159}

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

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

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

Step (7) : Rank the preference Order

Overall relative closeness and Rank of alternatives is shown in Table 10

Table 10

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

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

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 decision-makers 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

Table 14

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}

For negative ideal solution is shown in Table 16

Table 16 V1 = {0.136,0.114,0.140}

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

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

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

Step (7) : Rank the preference Order

Overall relative closeness and Rank of alternatives is shown in Table 20

Table 20

R1 > R3= R5 > R2 > R4

V. CONCLUSION :

TOPSIS is one of the Multi-Criteria Decision-Making 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 :

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

5. 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. 889-899.

6. Majid Behzadian, Khanmohammadi,S, Otaghsara, Morteza Yazdani, Joshua Ignatius.(2012). A state-of the art survey of Topsis applications, Expert systems with Applications, An

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