The Cardinal TOPSIS via Grey Relational Grade

Kunli Wen

and Meili You

Department of Electrical Engineering Education, Chienkuo Technology University, Changhua, Taiwan

Department of General Education, Chienkuo Technology University, Changhua, Taiwan

Keywords: Soft computing, Technique for order preference by similarity to ideal solution, Globalization grey relational

grade, Objective weighting.

Abstract: Quite a lot of soft computing models have over the past few years been developed. They are developed to

meet different purposes and needs. These models, however, may not satisfy the technique for order

preference by similarity to ideal solution (TOPSIS). Aware of this phenomenon, the present study applied

globalization grey relational grade of the grey system to convert subjective weighting in the computing

process of technique for order preference by similarity to ideal solution into objective weighting. Data

analysis demonstrated that applying grey relational grade to cardinal technique for order preference by

similarity to ideal solution was not only rational but also could transfer ordinal answer into cardinal answer.

1 INTRODUCTION

The Technique for Order of Preference by Similarity

to Ideal Solution (TOPSIS), originally developed by

Hwang and Yoon in 1981, is a multi-criteria

decision analysis method. The fundamental

assumption of TOPSIS is that the criteria are either

monotonically increasing or decreasing. It is based

on the concept that the positive ideal solution is

composed of the best score in all criteria, given, for

instance, benefit as the maximal value and cost as

the minimum value. In contrast, the negative ideal

solution is composed of the worst score in all

criteria, given, for instance, benefit as the minimum

value and cost as the maximum value (Wen and

You, 2018). According to TOPSIS, a set of

alternatives is measured in terms of the Euclidean

norm to compare their closeness to the positive ideal

solution. TOPSIS is based on the concept that the

chosen alternative should have the shortest

geometric distance from the positive ideal solution

and the longest geometric distance from the negative

ideal solution. Such a method, according to Pi, can

prevent an alternative from being both the shortest

distance to the positive ideal solution and the

negative ideal solution on the one hand, and being

both the longest distance from the positive ideal

solution and the negative ideal solution on the other

hand (Pi, 2005).

A close observation of TOPSIS reveals that one

step of the TOPSIS utilizes subjective weighting for

analysis. Different weighting inevitably generates

different result. There has been research which

either focused on soft computing and environment

area such as grey entropy-TOPSIS method (Liu et

al., 2014), combined TOPSIS with grey relation to

decide the weighting of TOPSIS in contractor

selection (Zavadskas et al., 2010), compared fuzzy

AHP and fuzzy TOPSIS for road pavement

maintenance prioritization (Ouma et al., 2015), used

the AHP and TOPSIS approaches under fuzzy

environment (Shahab, 2016) or applied fuzzy

TOPSIS-TODIM hybrid method for green supplier

selection (Khamseh and Mahmoodi, 2014). (Qian et

al., 2009) was the only one study applying the grey

relation grade method to TOPSIS. The present study,

which was based on the research method mentioned

above, used cardinal grey relational grade to convert

subjective weighting into objective weighting (Wen,

2013). Section Two of this study discussed the

mathematical model related to the Technique for the

Order Preference by Similarity to Ideal Solution.

The third section investigated and ranked four kinds

of drinking water in Changhua County as to their

quality by using this soft computing method. The

last part of this paper provided research findings and

proposed suggestions for forthcoming research.

Wen, K. and You, M.

The Cardinal TOPSIS via Grey Relational Grade.

DOI: 10.5220/0008189002930296

In The Second International Conference on Materials Chemistry and Environmental Protection (MEEP 2018), pages 293-296

ISBN: 978-989-758-360-5

293

2 MATHEMATICS MODEL

The analysis procedure of TOPSIS and grey

relational grade is described step by step (Wen,

2016).

2.1 TOPSIS

1. Input the project data















mnmmm

xxxx







321

3333231

2232221

1131211

(1)

2. Normalize the data in equation (1)





(2)

then show normalization matrix in equation (3)















mnmmm

rrrr







321

3333231

2232221

1131211

(3)

3. Decide the weighting of

],,,,[

321 ni





4. Calculate the weighting decision matrix





























mnmmm

mnnmmm

vvvv

rrrr











321

3333231

2232221

1131211

332211

3333322311

2233222211

1133122111





(4)

5. Calculate the positive ideal solution



and

ideal negative solution



),,,,(}.{max

321





vvvvvA 

),,,,(}.{min

321





vvvvvA 

(5)

6. Calculate the positive ideal distance



and

negative ideal distance











jiji

vvS

)(









jiji

vvS

)(

(6)

7. Calculate the relative approaching of ideal

distance

, then, weighting can be found.

njni

,,3,2,1,,,3,2,1 ,  









(7)

2.2 Objective Weighting Analysis

Step 3 of TOPSIS is observed to be subjectively

postulated. As for the existent data, a mathematical

method can be used to convert subjective weighting

into objective weighting. The present paper, which

applied grey relational grade to TOPSIS, could yield

objective result. The basic concept of grey relational

grade is shown below. Five kinds of globalization

grey relational grade have been developed in the

past few years(Wen, 2016). This study referred to

the mathematical method proposed by Liu.

)(

)]([1











(8)

where:

Ijnkmi

 ,,,3,2,1 ,,,3,2,1 

: Reference sequence,

: Inspected sequences

ii.

||)()(|| kxkx

jiij



According to Saaty, the eigenvector method can

be used to rank the sequence, and then choose an

optimal one.

1. Base on the original sequences

))(,,)3(,)2(,)1((

.............................................

))(,,)3(,)2(,)1((

33333

22222

11111

kxxxxx

mmmmm



(9)

2. Constructing the relative weighting matrix



][

, by using the cardinal globalization grey

relational grade method to find the grey relational

grade, which is called grey relational matrix.























mmmm







22221

11211

(10)

3. Find the eigenvalue for the relative weighting

matrix



][

RAR





4. Use eigenvector method to find the weighting

for each target





RPP

},{

,.....3,21 n

diag



5. The maximum



corresponding eigenvector is

the weighting for the sequence.

MEEP 2018 - The Second International Conference on Materials Chemistry and Environmental Protection

294

3 REAL EXAMPLE

Four kinds of running water, including tap water,

Funyuan spring water, Puli spring water and

Hungmaojing water, served as the objects for

investigation. The original data was based on the test

conducted by the Environmental Protection Bureau

of Changhua County, and was modified for scaling.

The analysis steps are shown below.

1. Table 1 shows the measurement results.

2. Table 2 shows the normalization of four kinds

of drinking water by using equation (2).

3. Table 3, which is based on the data in Table 2,

demonstrates the weighting of four kinds of water by

using the grey relational grade.

4. Table 4, which is based on the data in Table 3,

shows the normalization matrix of four kinds of

water by using equation (4).

5. Table 5 depicts the calculation of the positive

ideal solution and ideal negative solution by using

equation (6).

6. Table 6 demonstrates the calculation of the

relative approaching of ideal distance of four kinds

of water by using equation (7).

Table 1: The modified data of four kinds of drinking water.

Item/source

1. Turbidity(10 times)

7.0

17.0

3.0

1.0

2. pH

8.0

7.5

6.8

6.7

3.Chlorine (100mg/10l)

54.0

12.0

6.0

61.0

4. Sulfates (100mg/10l)

8.5

2.1

2.0

18.2

5. Free chlorine

1.0

23.0

0.5

7.0

6. Total hardness (100mg/10l)

24.2

10.4

6.8

33.4

7. Iron content (times 10)

4.0

2.0

1.0

14.0

8. Total number of viable cells (1,000mg/100l)

74.8

6.24

7.3

Table 2: The normalization results.

Item/source

1. Turbidity(10 times)

0.0018

0.0025

0.0165

0.0002

2. pH

0.0021

0.0011

0.0374

0.0012

3.Chlorine (100mg/10l)

0.0139

0.0018

0.0330

0.0111

4. Sulfates (100mg/10l)

0.0022

0.0003

0.0110

0.0033

5. Free chlorine

0.0003

0.0034

0.0028

0.0013

6. Total hardness (100mg/10l)

0.0062

0.0015

0.0374

0.0061

7. Iron content (times 10)

0.0010

0.0003

0.0055

0.0025

8. Total number of viable cells (1,000mg/100l)

0.0034

0.0111

0.0343

0.0013

Table 3: The weighting from Liu’s grey relational grade.

Water source

Weighting

0.6941

0.1226

0.1705

0.6885

Table 4: The normalization decision matrix after add the weighting.

Item/source

1. Turbidity(10 times)

0.0013

0.0003

0.0028

0.0001

2. pH

0.0014

0.0001

0.0064

0.0008

3.Chlorine (100mg/10l)

0.0097

0.0002

0.0056

0.0076

4. Sulfates (100mg/10l)

0.0015

0.0000

0.0019

0.0023

5. Free chlorine

0.0002

0.0004

0.0005

0.0009

6. Total hardness (100mg/10l)

0.0043

0.0002

0.0064

0.0042

7. Iron content (times 10)

0.0007

0.0000

0.0009

0.0017

8. Total number of viable cell (1,000mg/100l)

0.0023

0.0014

0.0059

0.0009

The Cardinal TOPSIS via Grey Relational Grade

295

Table 5: The positive ideal solution and ideal negative solution of four kinds of drinking water.

Water source



0.0214

0.0031

0.0099

0.0164



0.0108

0.0014

0.0115

0.0090

Table 6: The relative approaching of ideal distance of four kinds of drinking water.

Weighting/water

Ideal distance

0.3354

0.3111

0.5374

0.3543

Rank

*A: Tap water(Changhua). B: Funyuan spring water. C: Puli spring water. D: Hungmaojing water

4 CONCLUSIONS

The main purpose of the TOPSIS is to obtain the

sorting of all alternatives. Despite the effort of this

research to avoid the difficulty in comparing the

distance between two directions in the weighting

steps, it was subjective to a certain degree. The

research result could therefore be uncertain. One of

the major contributions of this study was using the

grey relational grade to convert subjective weighting

into objective weighting for further cardinal. The

present research, which referred to and analysis four

kinds of running water, effectively verified and

supported this soft computing method; The results

obtained from the cardinal TOPSIS were consistent

with the real situation.

Forthcoming study is suggested to implement

other cardinal weighting methods such as grey

cluster analysis and GM(h,N) to make analysis more

reliable.

ACKNOWLEDGEMENTS

The authors would to thank Taiwan Kansei

information association, for the supporting of the

toolbox to verify the data calculation.

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