Decision making is the process of selecting

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1 Jornal of Advances in Compter Engineering and Technology, (4) 06 A New Mlti-Criteria Decision Making Based on Fzzy- Topsis Theory Leila Yahyaie Dizaji, Sohrab khanmohammadi Received (05-09-) Accepted (06--) Abstract In this paper, a new extended method of mlti criteria decision making based on fzzy-topsis theory is introdced. Mostly, it is not possible to gather precise data, so decision making based on these data loses its efficiency. The fzzy theory has been sed to overcome this draw back. In mlti-criteria decision making, criteria can correlate with each other, most of which are ignored in classic MCDM. In this paper, correlation coefficient of fzzy criteria has been stdied to adapt the interrelation between criteria and a new algorithm is proposed to obtain decision making. Finally the efficiency of sggested method is demonstrated with an example. Key words - MCDM, correlation, fzzy-topsis. - Department of Compter Salmas Branch, Islamic Azad University, Salmas, Iran. (l.yahyaie@iasalmas.ac.ir) - Department of Compter Engineering, University of Tabriz,Tabriz,iran. I. INTRODUCTION Decision making is the process of selecting the most appropriate choice among many others. One of the main branches of decision making science is mlti criteria decision making (MCDM). In MCDM more than one criterion is important for the best choice. These criteria can be qalitative, qantitative, and positive or negative [-4]. When it s hard or impossible to get precise data, fzzy theory can be sed as an appropriate and strong tool for analyzing ambigos and imprecise problems [5]. One of the most common ways of MCDM is Topsis. Topsis is clear and nderstandable with no complexity. In Topsis, criteria weights and choice efficiencies shold be precise bt in practice it s not so. Therefore, most of the researchers try to apply fzzy data in Topsis. In most fzzy Topsis methods, some fzzy 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 fzzy data. Mahdavi et al. [7] has proposed fzzy- Topsis with transforming fzzy data to non-fzzy data. Wang and Elhag [8], Ding and Cho [9] have generalized fzzy-topsis on the basis of. Chen has solved fzzy-topsis method in one interval one leniency redction in [0]. Ye [] extended Topsis with interval nmbers. Three important phases in all MCDM methods for ranking are as follow:. Determining criteria and different choices.. Attribting Prices to weights of criteria and determining choices rate in proportion to different criteria. 3. Processing nmerical Prices for determining

2 40 Jornal of Advances in Compter Engineering and Technology, (4) 06 ranks of choices. Most researchers emphasize on the second and third stages bt the first one has not attracted mch attention. Sometimes srveying to choose an appropriate criterion is ignored and it becomes optional. Some researchers select inter related criteria and so it leads to nmeros criteria. Moreover, it becomes boring and hard to analyze the criteria becase of repetitive evalations and a lot of comparisons. So, correlation coefficient between variables is also stdied to remove variables with high inter relation amonts. Chadhr and Bhattachary [] have sed Spearman correlation coefficient for calclating correlation of two fzzy series. Hng and W [3] have applied Centroid method to calclate correlation of two fzzy series. They have shown that, these relations can be positive or negative. Urdakl and Tarselic [4] have sed Spearman coefficient for crisp nmbers. In this paper a new method of fzzy-topsis is proposed where fzzy nmbers are ranked directly. It also has been sed to find the correlation between variables and to remove highly related vales. Next sections are as follows: section incldes primary definition of sbject. Section3 describes correlation coefficients between criteria and section 4 incldes sggested algorithm. The efficiency of sggested method is demonstrated in section 5 by means of an experimental example. II. PRIMARY DEFINITIONS Topsis method has been developed by Wang and Lee [4]. Its rle is sch that the selected choice has the least distance of positive ideal soltion and the farthest from negative ideal soltion [, ]. Mlti criteria decision making (MCDM) methods have the benefit of evalating different choices. They can also analyze and evalate qalitative and qantitative criteria at the same time. A MCDM problem can be smmarized in a decision matrix as shown in Fig.. A A A m C xxxx xxxx xxxx m C xxxx xxxx xxxx m WWWW = wwww, wwww, wwww nnnn Fig. Decision matrix C n xxxx n xxxx n xxxx mn 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 prpose of decision making mechanism is selecting the best choice from alternatives,, sch that the selected alternative has the highest rank and efficiency. The introdced method is fzzy stated with fzzy nmerical choices. Definition. (Trianglar fzzy nmber): fzzy nmber A is referred as of crisp nmbers with (a<b<c).the membership fnction of trianglar fzzy nmber is defined as below: x a if a x b b a x c µ A( x) = if b x c () b c 0 otherwise Definition. A fzzy nmber A is called a positive fzzy nmber if µ ( x) = A 0 for all x < 0 Definition3. If A is a trianglar fzzy nmber L U and [ A] > 0 and [ A] for [0,], then is called a normalized positive trianglar fzzy nmber. A = A Note. If [ ] [ ] L, A U, then by choosing we can identify the center vale of A, and by we can identify the left and right

3 Jornal of Advances in Compter Engineering and Technology, (4) 06 4 extension of A. III. CORRELATION COEFFICIENT Measring 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 stdent s math grade and statistics grade. In this paper we ll se fzzy nmbers presented in [6] to calclate the correlation coefficients. In [6] credibility theory has been sed to calclate correlation coefficient between two trianglar nmbers. Credibility theory is a branch of mathematics sed to stdy behaviors of fzzy nmbers. If and are two trianglar fzzy variables, then correlation coefficient between N and M can be find by the following formla. ( b + a)( b + a) + ( c + b )( c + b ) pc r = + ( b + a ) + ( c + b ) ( b + a ) + ( c b ) () Where represents the correlation coefficient measres that have high correlation and are gained by the method presented in [3]. The pairs with correlation vales more than.8 or less than -.8 are affiliated measres and are eliminate by following steps. Calclate the correlation of a measre relating to all other measres as pairs. List the criteria in colmns and rows of correlation matrix. 3-The correlation coefficient is a measre 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 crrent pair are omitted. 4- Repeat step 3 for all pairs of matrix. SUGGESTED FUZZY ALGORITHM The new algorithm for extending Topsis method in a fzzy environment is as follows: Step: Determine evalation criteria Step4: Calclate correlation coefficient between criteria s to remove dependent criteria sing weights of criteria and the method proposed in [3]. Step5: Constrct fzzy decision matrix. We assme that the fzzy decision vale for each is a trianglar fzzy nmber. Step6: Calclate the normalized fzzy decision matrix. At first, for each fzzy nmber we calclate the set of -ct as: Therefore each frame is converted to a fzzy nmber by the proposed method in [9] which can be normalized as: l [ xij ] l [ n ij ] = i =,..., m, j =,... n m l (([ x ([ ij ] ) + xij ] ) ) i= [ xij ] [ n ij ] = i =,..., m, j =,... n m l (([ xij ([ ] ) + xij ] ) ) i= (3) (4) Now interval is a normal range of the interval. According to note we can transform this normalized interval in to a fzzy nmber sch as, when we obtain, and when we have: Then: Step: Determine weight (attribte importance) of criteria Step3: Determine decision alternatives

4 4 Jornal of Advances in Compter Engineering and Technology, (4) 06 is a normalized positive trianglar fzzy nmber corresponding to. Step7: Calclate nonsocial weighted matrix. ( v ij ) n m V = Vij = Nij. W j i =,... n, j =,..., m ) Where W j is the weight of criterion. [6, 7] (5) and Step8: The largest trianglar fzzy nmber and the smallest one are calclated for each colmn of nonsocial weighted matrix. For finding the biggest and the smallest fzzy nmber we apply the following relations proposed for trapezoidal nmbers [0]. For each linear ranking fnction R we have b a b R Also, if a R if and only if 0 a R b and c d, then: R a + c R b + d Therefore, we have the following linear ranking: l a = ( a, a, a, β ) l R( a ) = cl a + c a + c. + cβ. β Where C β, C, C, CL are constant nmbers which at least one of them is nonzero Hence, if l b = ( b, b, γ, θ ) then, a R b if and only if L L a + a + ( β ) b + b + ( θ γ ) So for ranking Eq.s we have: l L R( a) = a + a ( β ) b + b + ( θ γ ) If we se trianglar fzzy nmber = β, then we have: [0] l ( b b ) ( (6) L R a) = ( a + a ) + Step9: Calclate the ideal soltion and negative ideal soltion for each alternative. + A = A = { } { + + = = + (maxv j J ) i,,... m V, V,..., V, V + } i ij j n (7) { ij j J i= m} = { V V V j V n } V ( min ),,...,,...,, i (8) Step0: Find Eclidean distance of two trianglar fzzy nmbers as [8, 9, 5] The i-choice distance with ideals by sing Eclidean method is: L + L + ([ V ] [ V ] ) + ([ V ] [ V i ] ) )) m + + di = ( Vij, V j ) = ( ij i ij j= ( L L ] ) ([ [ V ] [ V i + V ] [ V ] ) )) m di = d( Vij, V j ) = ( ij ij i j= (9) (0) Step: Define proportional similarity of Aj with ideal soltion as below: di R i = d + d + i i () Step: Select the decision choice with larger. [6, 7] We show the efficiency of sggested algorithm with an illstrative edcational example. Illstrative example: University Professor s Ranking: First, we defined fifteen criteria, then these criteria were analyzed by Payamenoor niversity stdents and the collected data were sed to rank the three professors of the niversity. Step: Fifteen criteria are defined for evalation as listed in Table.

5 43 Jornal of Advances in Compter Engineering and Technology, (4) 06 C C C 3 C 4 C 5 C 6 C 7 C 8 C 9 C 0 C C C 3 C 4 C 5 TABLE. MEASUREMENT CRITERIA Does she/he have enogh knowledge of the sbject matter? Does she/he se new scientific findings related to the sbject matter? Does she/he answer the stdents qestions? Does she/he have the ability to state matters clearly? Does she/he se existing facilities (board, text book, pictre, chart, overhead)? Does she/he state clearly the objectives and sbjects of the beginning of class and have cohesion or does she/he teaches as a lesson plan? Does she/he perform in discssion with stdents or does she/he try to improve their intellectal prodctivity? Does she/he manage the class well? Does she/he se evalation throgh edcational term? Midterm exams, solving exercises, homework and project? Does she/he encorage and motivate stdents to stdy and research? Does she/he pay attention to stdent s logical sggestions and critics? Is there availability to teacher at niversity to ask qestions? Does she/he obey cltral and reciprocal respect? Is he/she on time and does she/he se the class time effectively? Is she/he interested in teaching? TABLE. PROPORTIONAL IMPORTANCE OF CRITERIA

6 44 Jornal of Advances in Compter Engineering and Technology, (4) 06 TABLE 3. THE CORRELATION COEFFICIENTS FOR EACH PAIR OF CRITERIA ciritera par r ciritera par r ciritera par r ciritera par r c -c c 3 -c c 5 -c c 8 -c 3 c -c c 3 -c c 5 -c c 8 -c 4 c -c c 3 -c c 5 -c c 8 -c c -c c 3 -c c 5 -c 3 c 9 -c c -c c 3 -c c 5 -c 4 c 9 -c c -c c 3 -c c 5 -c c 9 -c c -c c 3 -c c 6 -c c 9 -c 3 c -c c 3 -c c 6 -c c 9 -c 4 c -c c 3 -c c 6 -c c 9 -c c -c c 3 -c c 6 -c c 0 -c c -c c 3 -c c 6 -c c 0 -c c -c c 3 -c c 6 -c c 0 -c c -c c 4 -c c 6 -c c 0 -c c -c c 4 -c c 6 -c c 0 -c c -c c 4 -c c 6 -c c -c c -c c 4 -c c 7 -c 8 c -c 4 c -c c 4 -c c 7 -c 9 c -c 5 c -c c 4 -c c 7 -c c -c c -c c 4 -c c 7 -c c -c c -c c 4 -c c 7 -c c -c c -c c 4 -c c 7 -c 3 c -c c -c c 4 -c c 7 -c 4 c 3 -c 4 c -c c 4 -c c 7 -c c 3 -c c -c c 5 -c c 8 -c 9 c 4 -c c -c c 5 -c 7 c 8 -c c -c c 5 -c 8 c 8 -c c -c c 5 -c 9 c 8 -c

7 45 Jornal of Advances in Compter Engineering and Technology, (4) 06 C C C 3 C 4 C 5 C 6 C 7 C 8 C 9 TABLE 4. THE CORRELATION BETWEEN THE TWO SETS FOR CRITERIA IS MARKED WITH C C C 3 C 4 C 5 C 6 C 7 C 8 C 9 C 0 C C C 3 C 4 C 5 C 0 C C C 3 C 4 TABLE 5. 3 SELECTED CRITERIA S AFTER CALCULATING CORRELATION COEFFICIENT TABLE 6. DECISION MATRIX FOR 3 CRITERIA TABLE 7. NONSOCIAL WEIGHTED MATRIX AND THE LARGEST AND SMALLEST VALUE OF EACH COLUMN

8 Jornal of Advances in Compter Engineering and Technology, (4) Step: There are three alternatives A, A, A3 (representing the first, second and third teachers) represented by nmber, and 3 in Table Step3: Define weights of criteria and 3 alternatives for decision. The reslts are shown in Table. Step4: We compte coefficient correlation for every pair of criteria by Eq. (). In Table 3, pairs of criteria and their coefficient correlations are shown. If we consider all criteria withot calclating correlation coefficient and eliminate the correlated criteria reslts will be as listed in Tables 9, 0 and. Reslts of Tables 8 and illstrate that the same ranking has been obtained for the criteria 3 and 5. The ranking vales of two methods are shown in Figre. The correlated criteria (with correlation coefficient more than 0.8) are shown in Table 4 marked by. For example criterion c is dependent to c, c3, c4, c6, c0, c and c5. So, criteria c, c3, c4, c6, c0, and c can be sbstitted by c. This procedre repeats for other criteria and finally three criteria c5, c, c3 remains for decision making which are shown in Table 5. Step5 and step6: New fzzy decision matrix is generated for the three remaining criteria as shown in Table 6. Step7: Generate Nonsocial weight matrix weighed by sing Eq. (5) as shown in Table 7 Step 8: Find the biggest and smallest trianglar fzzy nmbers for each colmn according to Eq. (6) as shown in Table 7 Step 9: Find positive soltion and negative soltions, according to Eq.s (7), (8). Step0: Compte Eclidean distance of two trianglar fzzy nmbers by sing Eq.s (9) and (0). The reslts are shown in colmns and in Table 8. Fig.. The ranking vales with and withot the removal of correlated criteria CONCLUSION In this paper a new extension of Topsis is introdced for fzzy mlti criteria decision making (MCDM). MCDM is sed as a soltion for parallel programs possessing that have series of qalitative and qantitative estimates to rank different alternatives. This method covers both certain data and sbjective jdgments. The correlation coefficient between criteria is calclated to redce the nmber of criteria. An experimental example is sed to show the efficiency of introdces procedre. Step and step : Find the priority list of alternatives based on Ri (), as shown in colmn 4 of Table 8, that is A>A>A3. TABLE 8. RANKING DIFFERENT ALTERNATIVES

9 Jornal of Advances in Compter Engineering and Technology, (4) Table 8. Ranking different alternatives TABLE 9. DECISION MATRIX FOR 5 CRITERIA TABLE 0. NONSOCIAL WEIGHTED MATRIX, THE LARGEST AND SMALLEST NUMBER OF EACH COLUMN FOR 5 CRITERIA TABLE. RANKING DIFFERENT CHOICES FOR 5 CRITERIA

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