A New Multi-Criteria Decision Making Based on Fuzzy- Topsis Theory
Subject Areas : Fuzzy SystemsLeila Yahyaie 1 , Sohrab Khanmohammadi 2
1 - Department of Computer, Islamic Azad University, Salmas Branch, Salmas, Iran.
2 - Department of Computer Engineering, University of Tabriz,Tabriz, Iran.
Keywords:
Abstract :
[1] G. O. Young, “Synthetic structure of industrial plastics (Book style with paper title and editor),” in Plastics, 2nd ed. vol. 3, J. Peters, Ed. New York: McGraw-Hill, 1964, pp. 15–64.
[2] M.J. Asgharpour, Multi criteria decision making, fourth ed., Tehran University Press (In Farsi), 2004, pp. 456.
[3] J. Jiang , Yu-Wang Chen, Ying-wu Chen, Ke-wei Yang, TOPSIS with fuzzy belief structure for group belief multiple criteria decision making, Expert Systems with Applications 38 (2011) 9400–9406
[4] B. Vahdani, S. M. Mousavi, R. Tavakkoli-Moghaddam, Group decision making based on novel fuzzy modified TOPSIS method, Applied Mathematical Modeling. 35 (2011) 4257–4269,
[5] Timothy J.Ross , Fuzzy logic with engineering applications, second Ed. John Wiley & Sons Ltd, The Atrium, Southern Gate, Chic Hester, England , 2004
[6] M. Izadikhah, Using the Hamming distance to extend TOPSIS in a fuzzy environment, Journal of Computational and Applied Mathematics. 231 (2009) 200-207
[7] I. Mahdavi, N. Mahdavi-Amiri, A. Heidarzade,R.Nourifar, Designing a model of fuzzy TOPSIS in multiple criteria decision making, Applied Mathematics and Computation. 206 (2008) 607–617
[8] Y. M. Wang, T. M. S. Elhag, Fuzzy TOPSIS method based on alpha level sets with an application to bridge risk assessment, Expert Systems with Applications. 31 (2006) 309–319,
[9] J. F. Ding and C. C. Chou, A fuzzy MCDM model of service performance for container ports, Scientific research and Essays. 6 (2011) 559-556
[10] Ting-Yu Chen, Interval-valued fuzzy TOPSIS method with leniency reduction and an experimental analysis, Applied Soft Computing. 11 (2011) 4591–4606
[11] B.B. Chaudhuri and A. Bhattacharya, On correlation between two fuzzy sets, Fuzzy Sets and Systems. 118 (2001) 447-456,
[12] W. Hung, J. Wu, Correlation of intuitionistic fuzzy sets by centroid method, Information Sciences. 144 (2002) 219–225
[13] M. Yurdakul, Y. Tansel İÇ. Application of correlation test to criteria selection for multi criteria decision making (MCDM), models.Int J AdvManufTechnol. 40( 2009) 403–412
[14] T. C. Wang, H. D. Lee, Developing A fuzzy TOPSIS approach based on subjective weights and objective weights, Expert Systems with Applications. 36 (2009) 8980–8985
[15] W. Pedrycz, F. Gomide , An introduction to fuzzy sets analysis and design, Prentice’ Hall of India ,2004
[16] V. S. Vaidyanathan, Correlation of Triangular Fuzzy Variables Using Credibility Theory. International journal of computational cognition. HTTP://WWW.IJCC.US, 8 (2010)
[17] F. HosseinzadehLotfi, T. Allahviranloo, M. AlimardaniJondabeh, A New Method for Complex Decision Making Based on TOPSIS for Complex Decision Making Problems with Fuzzy Data, Applied Mathematical Sciences. 1 (2007) 2981-2987
[18] Z. Yue, An extended TOPSIS for determining weights of decision makers with interval numbers, Knowledge-Based Systems 24 (2011) 146–153
[19] C. H. Yeh, Y. H. Chang, Modeling subjective evaluation for fuzzy group multicriteria decision making, European Journal of Operational Research. 194 (2009) 464–473
[20] N. Mahdavi-Amiri, S. H. Nasseri, A. Yazdani, Fuzzy primal simplex algorithm for solving fuzzy linear programming problem, Iranian Journal of Operation Researcher (IJOR). 1 (2009) 68-84.
F. Liguo, L. Yanhong, A New MCDM Method in Transmission Network Planning Based on Gray Correlation Degree and TOPSIS, in: Proceedings of the 27th Chinese Control Conference, Kunming,Yunnan, China, 2008, pp.462-467
4
Journal of Advances in Computer Engineering and Technology
A New Multi-Criteria Decision Making Based on Fuzzy- Topsis Theory
Leila Yahyaie.Dizaji1, Sohrab khanmohammadi2
Received (Day Month Year)
Revised (Day Month Year)
Accepted (Day Month Year)
Abstract— In this paper, a new extended method of multi criteria decision making based on fuzzy-Topsis theory is introduced. Mostly, it is not possible to gather precise data, so decision making based on these data loses its efficiency. The fuzzy theory has been used to overcome this draw back. In multi-criteria decision making, criteria can correlate with each other, most of which are ignored in classic MCDM. In this paper, correlation coefficient of fuzzy criteria has been studied to adapt the interrelation between criteria and a new algorithm is proposed to obtain decision making. Finally the efficiency of suggested method is demonstrated with an example.
Key words :MCDM, correlation, fuzzy-Topsis.
I.INTRODUCTION
T
Decision making is the process of selecting the most appropriate choice among many others. One of the main branches of decision making science is multi criteria decision making (MCDM). In MCDM more than one criterion is important for the best choice. These criteria can be qualitative, quantitative, and positive or negative [1-4]. When it's hard or impossible to get precise data, fuzzy theory can be used as an appropriate and strong tool for analyzing ambiguous and imprecise problems [5]. One of the most common ways of MCDM is Topsis. Topsis is clear and understandable with no complexity. In Topsis, criteria weights and choice efficiencies should be precise but in practice it's not so. Therefore, most of the researchers try to apply fuzzy data in Topsis. In most fuzzy Topsis methods, some fuzzy data has been definitely eliminated, so some information has been lost as well. Izadikhan [6] has developed Topsis method for decision making in one interval or fuzzy data. Mahdavi et al. [7] has proposed fuzzy- Topsis with transforming fuzzy data to non-fuzzy data. Wang and Elhag [8], Ding and Chou [9] have generalized fuzzy-Topsis on the basis of 
. Chen has solved fuzzy-Topsis method in one interval one leniency reduction in [10]. Yue [11] extended Topsis with interval numbers.
Three important phases in all MCDM methods for ranking are as follow:
1. Determining criteria and different choices.
2. Attributing Prices to weights of criteria and determining choices rate in proportion to different criteria.
3. Processing numerical Prices for determining ranks of choices.
Most researchers emphasize on the second and third stages but the first one has not attracted much attention. Sometimes surveying to choose an appropriate criterion is ignored and it becomes optional. Some researchers select inter related criteria and so it leads to numerous criteria. Moreover, it becomes boring and hard to analyze the criteria because of repetitive evaluations and a lot of comparisons. So, correlation coefficient between variables is also studied to remove variables with high inter relation amounts. Chaudhur and Bhattachary [12] have used Spearman correlation coefficient for calculating correlation of two fuzzy series. Hung and Wu [13] have applied Centroid method to calculate correlation of two fuzzy series. They have shown that, these relations can be positive or negative. Urdakul and Tarselic [14] have used Spearman coefficient for crisp numbers. In this paper a new method of fuzzy-Topsis is proposed where fuzzy numbers are ranked directly. It also has been used to find the correlation between variables and to remove highly related values. Next sections are as follows: section 2 includes primary definition of subject. Section3 describes correlation coefficients between criteria and section 4 includes suggested algorithm. The efficiency of suggested method is demonstrated in section 5 by means of an experimental example.
II. Primary definitions
Topsis method has been developed by Wang and Lee [14]. Its rule is such that the selected choice has the least distance of positive ideal solution and the farthest from negative ideal solution [1, 2]. Multi criteria decision making (MCDM) methods have the benefit of evaluating different choices. They can also analyze and evaluate qualitative and quantitative criteria at the same time. A MCDM problem can be summarized in a decision matrix as shown in Fig. 1.
Cn | … | C2 | C1 |
|
…
| … … … … |
…
|
…
| A1 A2 … Am |
Fig.1 Decision matrix
Where alternatives
are options and 
are decisions making Criteria. Utility of each choice regarding the criterion is referred by 
and 
is the weight (importance factor) of criterion
.The main purpose of decision making mechanism is selecting the best choice from alternatives, 
, such that the selected alternative has the highest rank and efficiency. The introduced method is fuzzy stated with fuzzy numerical choices.
Definition 1. (Triangular fuzzy number): fuzzy number A is referred as 
of crisp numbers with (a<b<c).The membership function of triangular fuzzy number is defined as below:
|
(1) |
Definition2: A fuzzy number 
is called a positive fuzzy number if
for all 
Definition3: If is a triangular fuzzy number and 
for
, then
is called a normalized positive triangular fuzzy number.
Note1: If
, then by choosing
we can identify the center value of
, and by 
we can identify the left and right extension of
.
III. Correlation Coefficient
Measuring correlation coefficient between two variables is important since it shows strength and rate of relationship between two variables. For example there is correlation between the student's math grade and statistics grade.
In this paper we'll use fuzzy numbers presented in [16] to calculate the correlation coefficients. In [16] credibility theory has been used to calculate correlation coefficient between two triangular numbers. Credibility theory is a branch of mathematics used to study behaviors of fuzzy numbers. If 
and 
are two triangular fuzzy variables, then correlation coefficient between N and M can be find by the following formula.
|
(2) |
Where 
represents the correlation coefficient measures that have high correlation and are gained by the method presented in [13]. The pairs with correlation values more than .8 or less than -.8 are affiliated measures and are eliminate by following steps.
Calculate the correlation of a measure relating to all other measures as pairs.
List the criteria in columns and rows of correlation matrix.
3-The correlation coefficient is a measure to compare the pair with all other pairs of matrix and comparing one pair of the correlation coefficient with all other pairs. All other pairs that are correlated with current pair are omitted.
4- Repeat step 3 for all pairs of matrix.
Suggested fuzzy algorithm
The new algorithm for extending Topsis method in a fuzzy environment is as follows:
Step1: Determine evaluation criteria
Step2: Determine weight (attribute importance) of criteria
Step3: Determine decision alternatives
Step4: Calculate correlation coefficient between criteria’s to remove dependent criteria using weights of criteria and the method proposed in [13].
Step5: Construct fuzzy decision matrix.
We assume that the fuzzy decision value for each 
is a triangular fuzzy number. 
Step 6: Calculate the normalized fuzzy decision matrix.
At first, for each fuzzy number we calculate the set of α-cut as:
Therefore each frame is converted to a fuzzy number by the proposed method in [9] which can be normalized as:
|
(3)
(4) |
Now interval
is a normal range of the interval
. According to note1 we can transform this normalized interval in to a fuzzy number such as
, when we obtain 
, and when we have:
|
Then:
|
is a normalized positive triangular fuzzy number corresponding to.
Step7: Calculate nonsocial weighted matrix.
|
(5) |
Where 
is the weight of criterion 
and
. [6, 17]
Step8: The largest triangular fuzzy number and the smallest one are calculated for each column of nonsocial weighted matrix. For finding the biggest and the smallest fuzzy number we apply the following relations proposed for trapezoidal numbers [10].
For each linear ranking function R we have
if and only if 
Also, if
and 
, then:
Therefore, we have the following linear ranking:
Where 
are constant numbers which at least one of them is nonzero Hence, if 
then, 
if and only if
So for ranking Eq.s we have:
If we use triangular fuzzy number
, then we have: [20]
| (6) |
Step9: Calculate the ideal solution and negative ideal solution for each alternative.
| (7)
(8) |
Step10: Find Euclidean distance of two triangular fuzzy numbers as [8, 9, 5]
The i-choice distance with ideals by using Euclidean method is:
| (9)
(10) |
Step11: Define proportional similarity of Aj with ideal solution as below:
| (11) |
Step12: Select the decision choice with larger
. [6, 17]
We show the efficiency of suggested algorithm with an
1Corresponding Author: Department of Computer salmas Branch, Islamic Azad
University, salmas, Iran, (l.yahyaie@iausalmas.ac.ir)
2 Department of Computer Engineering, University of Tabriz,Tabriz,iran
, (khan@tabrizu.ac.ir)
illustrative educational example.
Illustrative example:
University Professor’s Ranking:
First, we defined fifteen criteria, then these criteria were analyzed by Payamenoor university students and the collected data were used to rank the three professors of the university.
Step1: Fifteen criteria are defined for evaluation as listed in Table 1.
Table 1. Measurement criteria
Step2: There are three alternatives A1, A2, A3 (representing the first, second and third teachers) represented by number 1, 2 and 3 in Table
Step3: Define weights of criteria and 3 alternatives for decision. The results are shown in Table 2.
Table 2. Proportional importance of criteria
Step4: We compute coefficient correlation for every pair of criteria by Eq. (2). In Table 3, pairs of criteria and their coefficient correlations are shown.
Table 3. The correlation coefficients for each pair of criteria
The correlated criteria (with correlation coefficient more than 0.8) are shown in Table 4 marked by *.
For example criterion c1 is dependent to c2, c3, c4, c6, c10, c12 and c15. So, criteria c2, c3, c4, c6, c10, and c12 can be substituted by c1. This procedure repeats for other criteria and finally three criteria c5, c1, c13 remains for decision making which are shown in Table 5.
Table 4. The correlation between the two sets for criteria is marked with *
Table 5. 3 selected criteria's after calculating correlation coefficient.
Step5 and step6: New fuzzy decision matrix is generated for the three remaining criteria as shown in Table 6.
Table 6. Decision matrix for 3 criteria
Step7. Generate Nonsocial weight matrix weighed by using Eq. (5) as shown in Table 7
Table 7. Nonsocial weighted matrix and the largest and smallest value of each column
Step 8: Find the biggest and smallest triangular fuzzy numbers for each column according to Eq. (6) as shown in Table 7
Step 9: Find positive solution and negative solutions, according to Eq.s (7), (8).
Step10: Compute Euclidean distance of two triangular fuzzy numbers by using Eq.s (9) and (10). The results are shown in columns 1 and 2 in Table 8.
Step11 and step 12: Find the priority list of alternatives based on Ri (11), as shown in column 4 of Table 8, that is A1>A2>A3.
Table 8. Ranking different alternatives
If we consider all criteria without calculating correlation coefficient and eliminate the correlated criteria results will be as listed in Tables 9, 10 and 11. Results of Tables 8 and 11 illustrate that the same ranking has been obtained for the criteria 3 and 15.
Table 9. Decision matrix for 15 criteria
Table 10. Nonsocial weighted matrix, the largest and smallest number of each column for 15 criteria
Table 11. Ranking different choices for 15 criteria
The ranking values of two methods are shown in Figure 2.
Fig. 2. The ranking values with and without the removal of correlated criteria
Conclusion
In this paper a new extension of Topsis is introduced for fuzzy multi criteria decision making (MCDM). MCDM is used as a solution for parallel programs possessing that have series of qualitative and quantitative estimates to rank different alternatives. This method covers both certain data and subjective judgments. The correlation coefficient between criteria is calculated to reduce the number of criteria. An experimental example is used to show the efficiency of introduces procedure.
References
[1] G. O. Young, “Synthetic structure of industrial plastics (Book style with paper title and editor),” in Plastics, 2nd ed. vol. 3, J. Peters, Ed. New York: McGraw-Hill, 1964, pp. 15–64.
[2] M.J. Asgharpour, Multi criteria decision making, fourth ed., Tehran University Press (In Farsi), 2004, pp. 456.
[3] J. Jiang , Yu-Wang Chen, Ying-wu Chen, Ke-wei Yang, TOPSIS with fuzzy belief structure for group belief multiple criteria decision making, Expert Systems with Applications 38 (2011) 9400–9406
[4] B. Vahdani, S. M. Mousavi, R. Tavakkoli-Moghaddam, Group decision making based on novel fuzzy modified TOPSIS method, Applied Mathematical Modeling. 35 (2011) 4257–4269,
[5] Timothy J.Ross , Fuzzy logic with engineering applications, second Ed. John Wiley & Sons Ltd, The Atrium, Southern Gate, Chic Hester, England , 2004
[6] M. Izadikhah, Using the Hamming distance to extend TOPSIS in a fuzzy environment, Journal of Computational and Applied Mathematics. 231 (2009) 200-207
[7] I. Mahdavi, N. Mahdavi-Amiri, A. Heidarzade,R.Nourifar, Designing a model of fuzzy TOPSIS in multiple criteria decision making, Applied Mathematics and Computation. 206 (2008) 607–617
[8] Y. M. Wang, T. M. S. Elhag, Fuzzy TOPSIS method based on alpha level sets with an application to bridge risk assessment, Expert Systems with Applications. 31 (2006) 309–319,
[9] J. F. Ding and C. C. Chou, A fuzzy MCDM model of service performance for container ports, Scientific research and Essays. 6 (2011) 559-556
[10] Ting-Yu Chen, Interval-valued fuzzy TOPSIS method with leniency reduction and an experimental analysis, Applied Soft Computing. 11 (2011) 4591–4606
[11] B.B. Chaudhuri and A. Bhattacharya, On correlation between two fuzzy sets, Fuzzy Sets and Systems. 118 (2001) 447-456,
[12] W. Hung, J. Wu, Correlation of intuitionistic fuzzy sets by centroid method, Information Sciences. 144 (2002) 219–225
[13] M. Yurdakul, Y. Tansel İÇ. Application of correlation test to criteria selection for multi criteria decision making (MCDM), models.Int J AdvManufTechnol. 40( 2009) 403–412
[14] T. C. Wang, H. D. Lee, Developing A fuzzy TOPSIS approach based on subjective weights and objective weights, Expert Systems with Applications. 36 (2009) 8980–8985
[15] W. Pedrycz, F. Gomide , An introduction to fuzzy sets analysis and design, Prentice’ Hall of India ,2004
[16] V. S. Vaidyanathan, Correlation of Triangular Fuzzy Variables Using Credibility Theory.
International journal of computational cognition. HTTP://WWW.IJCC.US, 8 (2010)
[17] F. HosseinzadehLotfi, T. Allahviranloo, M. AlimardaniJondabeh, A New Method for Complex Decision Making Based on TOPSIS for Complex Decision Making Problems with Fuzzy Data, Applied Mathematical Sciences. 1 (2007) 2981-2987
[18] Z. Yue, An extended TOPSIS for determining weights of decision makers with interval numbers, Knowledge-Based Systems 24 (2011) 146–153
[19] C. H. Yeh, Y. H. Chang, Modeling subjective evaluation for fuzzy group multicriteria decision making, European Journal of Operational Research. 194 (2009) 464–473
[20] N. Mahdavi-Amiri, S. H. Nasseri, A. Yazdani, Fuzzy primal simplex algorithm for solving fuzzy linear programming problem, Iranian Journal of Operation Researcher (IJOR). 1 (2009) 68-84.
F. Liguo, L. Yanhong, A New MCDM Method in Transmission Network Planning Based on Gray Correlation Degree and TOPSIS, in: Proceedings of the 27th Chinese Control Conference, Kunming,Yunnan, China, 2008, pp.462-467
Leila yahyaie received the B.sc. degree in computer
Software engineering from the Islamic azad university Tabriz branch, Iran in 2001.she also received M.sc degree in computer software engineering from Islamic Azad University Shabestar branch. She is currency ph.d student in Science and Research Branch of Islamic Azad
University
Sohrab Khanmohamadi received his BA
From Sharif University of Technology, Iran, 1977.He received ms degree from ENSAE in field of Adavanced Autmatic and System, France in 1980 and the Ph.D. degree in Automatic from ENSAE ,Frace, 1983, currency he is Professor of Faculty of Electrical & Computer Engineering in Tabriz University
