fuzzy simple additive weighting method by

Intelligent Automation and Soft Computing, Vol. 11, No. 4, pp. 235-244, 2005 Copyright © 2005, TSI® Press Printed in the USA. All rights reserved

FUZZY SIMPLE ADDITIVE WEIGHTING METHOD BY PREFERENCE RATIO M. MODARRES Sharif University of Technology [email protected] AND

S. SADI-NEZHAD Industrial Management Institute [email protected] Tehran, Iran

ABSTRACT—Although simple additive weighting method (SAW) is the most popular approach for classical multiple attribute decision making (MADM), it is not practical any more if information is fuzzy. The existing methods of Fuzzy Simple Additive Weighting method (FSAW) apply defuzzification which distorts fuzzy numbers. Furthermore, most of the methods usually require lengthy and laborious manipulations. In this paper, we develop a new a fuzzy simple additive weighting method for multiple attribute decision making problems. To avoid defuzzification round off errors caused by multiplication or other arithmetic manipulations, fuzzy numbers are ranked prior to any fuzzy arithmetic in this method. The ranking is done by preference ratio concept which compares fuzzy numbers pairwise. Since this method eventually assigns crisp scores to alternatives, it is practical and more realistic. We also present three other algorithms for ranking or normalizing fuzzy numbers which are actually sub algorithms of the main method. Key Words: FSAW, MADM, Fuzzy numbers, Preference ratio, Decision making, Ranking.

1. INTRODUCTION Due to the nature of real world problems, the collected data usually involve some kind of uncertainty. As a matter of fact, many pieces of information can not be quantified due to their nature. Incomplete information or partial ignorance are also some other causes of resorting to fuzziness. In many cases although exact information can be obtained, some approximated information deemed to be good enough to avoid high expense of precise data gathering. Therefore, many researchers prefer to incorporate fuzzy information into their decision making models in order to produce more realistic results. Furthermore, to some researchers, information arising from human mental phenomena can be expressed more realistically by fuzzy numbers in comparison with crisp or even random numbers. There are a variety of fuzzy decision models available. In general, they can be divided into two main groups, multiple attribute decision making (MADM) and multi-objective decision making (MODM) models. By adopting MADM approach, the decision maker selects among a finite set of alternatives, where each alternative is also evaluated by more than one attribute. These attributes usually are in conflict with each other and they are different from the point of view of their importance to the decision maker. Due to simplicity and practicality, Simple Additive Weighting (SAW) is the most popular method of classical MADM. In this method alternatives are measured by some attributes. Then, each alternative is assigned a score which is the weighted sum of these attributes. However, this method does not work easily when information is not crisp any more. Complexity of the existing fuzzy simple additive weighting (FSAW) methods arises from normalizing and ranking fuzzy numbers.

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In this paper, we develop a new method for solving fuzzy MADM problems within the framework of simple additive weighting (SAW) approach. What makes this method distinguished from the others, is its approach to defuzzyfication. The existing methods of solving various fuzzy decision making models of MADM calculate the score of alternatives by fuzzy arithmetic first and compare them by a fuzzy ranking technique. The main difficulty of this kind of approach is the distortion of triangular or trapezoidal numbers by fuzzy arithmetic. For example, multiplying two triangular fuzzy numbers A and B results in a fuzzy number C, which is not a triangular number any more, see Lai and Hwang [13]. However, in the existing methods the output of multiplication of two triangular fuzzy numbers is approximated to be another triangular fuzzy number. Naturally, when the number of fuzzy multiplication or other fuzzy arithmetic increases the round off errors accumulate and consequently the results are far from being realistic. To overcome this difficulty, the comparison is done prior to any fuzzy arithmetic in the proposed method. Clearly, an appropriate ranking method is needed for this approach. As we will show in the next section, ranking method by Preference Ratio concept preserves the characters of fuzzy numbers. Thus, we apply this approach in the proposed method. However, a new concept of Equivalence is also required to apply this technique, which is also introduced by considering the definition of preference ratio. By applying preference ratio, we will show the proposed method works fast and ends up with crisp score for each alternative while the existing methods usually end up with fuzzy score. To see the general concept of fuzzy equivalence, the reader is referred to Dubois and Prade [7]. Hwang and Yoon [8] reviewed the classical MADM methods. To get acquainted with fuzzy simple additives weighting method, the reader is referred to Chen and Hwang [3]. On the basis of their classification, the methods are divided into two categories, α-cut and fuzzy arithmetic. Bass and Kwakernaak [1] suggested a method of α-cut category, in which a value for µUi(ui) is determined for an assigned α1 ≤ 1 first. On this basis, the value of each alternative with respect to attributes, as well as the importance weights of attributes are estimated. Then, a lower and an upper bound for ui is calculated. When the size of a problem increases to more than ten attributes and ten weights, this method is not applicable due to the tremendous burden of over a million ui values calculation. Later, Kwakernaak [9] modified Baas and Kwakernaak's method by developing a modified algorithm. Although the basis of this algorithm is the same as before, it is more efficient. Dubois and Prade [6] also developed an algorithm on the basis of α-cut technique. However, it was much more efficient than Baas and Kwakernaak's because it did not apply trial and error technique. Finally, Cheng and McInnis [4] suggested another algorithm by converting continuous membership function into discrete ones. This algorithm also uses α-cut technique. It is necessary to mention the final step in all of these methods is to rank the scores of the alternatives that are fuzzy numbers. Bonissone [2] approximated the information in terms of L-R type trapezoidal number and applied arithmetic operations on fuzzy numbers. This method was appropriate for fuzzy linguistic terms only and could not be applied for general fuzzy numbers directly. The main difference between this method and the previous one is that it is not using the α-cut technique. Dong and Wang [5] proposed an algorithm to capture fuzzy weighted average of fuzzy numbers. Then, Tian-Shyliou and Wang [11] modified it by α-cut method and made it more efficient to calculate the membership function as well as more accurate. Yuan [12] introduced four following criteria to evaluate fuzzy ranking methods. 1- Fuzzy preference representation. 2- Notion of ordering 3- Distinguishability 4- Robustness These four criteria help us to evaluate Preference ratio method. We can show this method works quite well, accordingly. Tseng and Klein [14] also defined equivalency for two fuzzy numbers, although it is different from ours. They also compare the alternatives at the end. In the next section, the concept of preference ratio is introduced. We also present an algorithm to rank continuous fuzzy numbers on the basis of preference ratio. Then, in section 3 we introduce the concept of equivalence for a pair of fuzzy numbers. To apply the definition, an algorithm is also introduced in this section. A new method is also presented in section 4 on the basis of preference ratio to rank and normalize

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fuzzy numbers. Finally, in section 5, we sum up the previous results to present a method to make decision by fuzzy simple additive weighting method.

2. PREFERENCE RATIO Our proposed approach is developed on the basis of preference ratio concept as well as ranking method introduced by Modarres and Sadi-Nezhad [10]. In that method, fuzzy numbers are evaluated point by point. Then, the overall preference over all points is calculated. As such, the preference is relative rather than absolute. Suppose the objective is to rank I fuzzy numbers. Let Ni be the ith one defined over a real domain Si ⊂ R, and is identified by a membership function ( µ N i (x) , x ∈ S i ), where µ N i ∈ [0,1]. Let Si be the support of Ni, i.e., Si ={x,

µN

I

i

(x) >0 },

Ω = ∪ S i , then Ω is the union of the support of all fuzzy numbers. In i =1

other word, fuzzy numbers are ranked over Ω . We assume the fuzzy numbers do not have disjoint spans, because in that case ranking is clear. A fuzzy number is evaluated by a function called Preference Function. At each point, α ∈ Ω , this function is defined as follows, U

∫ µ ( x)dx G (α ) = α ∫ µ ( x)dx U

,

(1)

L

where, µ(x) is the membership function of the fuzzy number, L = min{x : x ∈ Ω } and U=max{x: x ∈ Ω }. This function has the same definition as 1-F( α ) in probability theory, where F( α ) = P[X ≤ α ] is the distribution function. At α ∈ Ω , let p( α ) = i denote the ith fuzzy number which is the most preferred one. Therefore, p( α ) = i,

if

Gi( α )= max{Gj( α ), j ∈ I},

where, Gj( α ) is the preference function of jth fuzzy number. Let, Ω i be the set of points at which the ith number is ranked number one. Then,

Ω i = { α ∈ Ω, p( α )=i}.

(2)

Definition 1. For the ith fuzzy number, R(i), the Preference Ratio, is by definition, the percentage of

Ω that the ith fuzzy number is the most preferred one. Then, R(i) =

Ωi Ω

(3)

where, Ω i and Ω are respectively the lengths of the real set Ω i and Ω

2.1 Preference ratio for continuous numbers

To determine preference ratio for fuzzy numbers Ai, i = 1, …, n, it is required to calculate Gi( α ) from (1) at each point α ∈ Ω first, and for every i = 1, …, n. Then, Ω i is calculated from (2) and R(i) from (3). For special cases, simplified algorithms can be developed. For instance, in case of triangular fuzzy numbers (TFN), which are manipulated frequently in fuzzy literature, an algorithm is developed by Modarres and Sadi-Nezhad [10]. In this section we present an algorithm on the basis of search technique for continuous fuzzy numbers. Although, the following algorithm is a search technique, it is not a laborious on computers and the results are obtained relatively fast.

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Algorithm I Let A1 and A2 be two fuzzy numbers with continuous membership functions and joint spans. Then, we calculate Gj(α), for j = 1, 2 at each interval of e in Ω where, e=

U −L 1000

(4)

and, L = min{x : x ∈ Ω }

U= max{x : x ∈ Ω }

Clearly, by setting smaller interval and naturally increasing the number of these intervals, more accurate results can be obtained. Step 1. Read A1 and A2 with their membership functions of µA1 and µA2. Step 2. Determine L and U as well as e from (4). Set α = L +e , R(A1) = 0, and R (A2) = 0. Step 3. Calculate Gj(α), for j = 1, 2. If G1(α) > G2(α), then set R (A1) = R (A1) +e and go to step 4. If G1(α) < G2(α), then set R (A2) = R (A2) +e and go to step 4. If G1(α) = G2(α), then set R (A1) = R (A1) +e/2 and R (A2) = R (A2) +e/2 . Step 4. Set α = α +e. If α < U, then go to step 3. Otherwise stop.

3. EQUIVALENCE BY PREFERENCE RATIO If the objective is to rank more than two fuzzy numbers, then comparing them pair-wise is not a practical approach especially if there are too many fuzzy numbers to rank. In this case, one practical approach is to compare each fuzzy number with a specified one as a bench mark. This way, a ratio is obtained for each fuzzy number. Then, rank them according to those ratios. Clearly to apply this approach, an efficient ranking method is necessary. By applying preference ratio concept, comparing any pair of fuzzy numbers end up with a crisp ratio. To do that, we change the scale of span of fuzzy numbers. In other words, we multiply a fuzzy number A by a real number k. Definition 2. a) Two fuzzy numbers A and B are equal by preference ratio criteria if R(A) =R(B) =0.5, where R(A) and R(B) are preference ratio of A and B, respectively. Equivalence by preference ratio is shown as follows. PR

A≡B

(5)

PR

b) If kA ≡ B,$ then we say k is the equivalence multiplier of A with respect to B. Note: In the remaining parts of this paper, when we say two fuzzy numbers are equal we mean they are equal by preference ratio criteria. Lemma 1. Let k ≠ 0 be a real number and A and B two fuzzy numbers. a) If A and kB are equal (by preference ratio criteria), so are

1 A and B. k

k1 > k2, then R(k1 A) \ ≥ R(k2 A). Proof: One should note that geometrically speaking kA has the same shape as A but shifted to right or left, depending on the value of k. (If k >1, then kA is on the left of A.) Thus, µ (x), L, α for fuzzy number A is equal to µ (kx), kL, k α , respectively for kA. b) If

a. Clearly from (1), GA(α) and GkA(kα) are equal. It means the preference function is invariant when the scale changes. Therefore, if both A and kB are multiplied by 1/k, then equality still holds. b. Let x ∈ Ω . As discussed above,

x

G k A (x) = G A( k ) , i i

i = 1,2. On the other hand, from (1), G( α 1)

≤ G( α 2) if α 1 ≥ α 2. Thus, Since k1 > k2, then G k A (x) > G k A (x), ∀x ∈ Ω . Then, from (3) R(k1 i i A) \ ≥ R(k2 A).

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3.1 Making a pair of fuzzy numbers equivalence To make two fuzzy numbers A and B equivalent (by preference ratio criteria) it is necessary to find a real number k such that (5) holds for kA and B. Here, we present an algorithm to obtain k, the equivalence multiplier. This algorithm is developed on the basis of search techniques. Since the number of iterations in any search technique heavily depends on the search range, it is necessary to make this range as short as possible. In other words, obtaining a reasonable upper bound k and a lower bound k for k avoids extra laborious manipulations. It is obvious when two fuzzy numbers are equal the intersection of their spans can not be empty. In figure 1 (a) R(N1) and R(N2) can not be equal because from (3) R(N1) = 0 while R(N2)=1. However, in (b) the equivalence of N3 and N4 is possible. In other words, a necessary condition for equivalence of two fuzzy numbers is the overlap their spans.

Figure 1. Positions of two equal or unequal fuzzy numbers. PR

Theorem 1 Let N1 ≥ 0 and N2 ≥ 0 be two fuzzy numbers and kN1 ≡ N2, with span of S1 and S2, respectively. Then, the upper and lower bound of k, is as follows. a) If R (N1) < R (N2), then,

U L k = maz{ 2 , 1}, and k = 2 U1 L1

(6)

b) If R (N2) < R (N1), then,

k=

L2 U . and k = maz{ 2 ,1}, U1 L1

(7)

where, Li = min{x: x ∈ Si} and Ui = max{x: x ∈ Si}, for i=1,2. Proof: a) Since two equivalent fuzzy numbers must have overlapping spans, then the following relation holds. kU1 ≥ L2. Consequently, k ≥ L2 / U1. On the other hand, since R(N2) < R(N1), then k >1. Therefore, the proof for k of part (6) is complete. The necessary condition of overlap of spans also results in kL1 ≤ U2 or k =

U2 . Similar argument also holds for part (b). L1

By applying the following algorithm, an equivalence multiplier can be obtained to make two fuzzy numbers equivalent.

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Algorithm II Let N1 and N2 be two non-negative fuzzy numbers. (In case N1 and N2 are negative, then by changing the scales we make them non-negative.) First, set δ as the desired measure of accuracy of the algorithm. Then, determine R(N1) and R(N2). Step 1. Find k and k from (6) or (7). Step 2. Calculate k = ( k + k )/2. Step 3. Determine R(k N1) and R(N2).

Step 4. If R(k N1) < δ -0.5, then set k = (k + k )/2 . Go to step 2. If R(k N1) > k)/2. Go to step 2. If |R(N1)- 0.5| < δ , stop.

δ

+ 0.5, then set k = (k +

4. RANKING AND NORMALIZING BY PREFERENCE RATIO As mentioned before different methods are developed for dealing with fuzzy numbers in FSAW. In our proposed method, fuzzy numbers are ranked by applying preference ratio technique first. It results in assigning a crisp score to each number. To do that, an algorithm is introduced to rank and if necessary to normalize them as well.

Algorithm III Let Ni for i =1, … , m be trapezoidal fuzzy numbers. They may represent the measurement of alternatives by an attribute or importance weight of the attributes. Let ith number be expressed as, Ni = [Li, M1i, M2i, Ui], i =1, … , m, where, Li, M1i, M2i and Ui are respectively x-coordinate of the vertices of the trapezoid. A triangular fuzzy number is a special case, where M1i, = M2i, for i=1, …, m. The objective is to rank the numbers in increasing or decreasing order. The procedure for either case is different but the main steps are the same. Here, we present the algorithm for the case when the objective is to rank them in decreasing order (determining the maximum). Then, we modify the algorithm for the other case.

~

Step 1. Define a new fuzzy number N such that,

~ N = [ L, M , M , U ] where, L = max {Li, i=1, … , m}, M = max {M1i, i=1, … , m}

M = max {M2i, i=1, … , m} and, U = max {Ui, i=1, … , m}.

~

If all Nis are TFNs, then clearly N will be a TFN too. PR

Step 2. To compare the fuzzy numbers find a crisp vector k =(k1, k2, …km), such that N ≡ ki N , for i=1,…,m. Step 3. To normalize vector k, find a vector W = ( W1, W2, … ,Wm), such that,

Wi =

k

∑

i m j =1

kj

,

i=1, …, m.

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241

Note: a. In case the objective is to rank the fuzzy numbers scores in increasing order (minimization case), then, two modifications are required. First, in step 1, all four quantity of L, M , M and M are the minimum PR

value of corresponding values. Second, in step 2, ki ni ≡ N , for i = 1,…,m b. If fuzzy numbers represent weights, then they should be normalized. However, if they represent attributes then they need to be ranked only. Therefore, it is not necessary to normalize them. In this case, step 3 is not required any more. Example 1 Let importance weight of three attributes be expressed in linguistic terms as medium low, medium high and high. To rank and normalize them by crisp numbers we follow algorithm III. However, first it is necessary to express linguistic terms into fuzzy numbers by determining membership functions. First, the linguistic fuzzy numbers are expressed as membership function, illustrated in Figure 2.

Figure 2. Scaling system for linguistic terms.

These three numbers are converted into triangular fuzzy numbers (TFN) as follows. A1 = (0.6, 0.8, 1), A2 =(0.5, 0.65, 0.8), A3=(0.2, 0.35, 0.5).

~

Step 1. Define a TFN, A = ( L, M , M , U ) = (0.6,0.8,0.8,1). Step 2. By applying algorithm II, these numbers are to compared. Then, k1= 1, k2 = 0.538, and k3 = 0.437. Step 3. Since these fuzzy numbers are weights then the vector of k, should be normalized. Thus, W = (W1, W2, W3) = (0.506, 0.272, 0.222). Example 2 One attribute to select an assembly line from five suppliers is the satisfaction of the former clients of these suppliers. In Table I, each alternative is represented by a trapezoidal fuzzy number.

~

In step 1, we define N whose elements are the maximum value of corresponding elements of fuzzy

~

~

numbers. Then, N = (12, 13, 13, 18), happens to be a TFN. In step 2, find ki, such that Ai = k i A By applying algorithm I, vector k = (k1, k2, k3, k4, k5) = (0.7557, 0.6361, 0.9434, 0.8834, 0.7616) is obtained. Since we are only concerned with ranking of these five fuzzy numbers and not normalizing them, then it is not necessary to apply step 3 of algorithm II. Table I. Client Satisfaction for alternatives Alternatives j 1 2 3 4 5

Lj 7 5 10 12 5

Customer Satisfaction M1j M2j 10 12 11 11 12 12 13 13 7 12

Uj 14 12 18 14 17

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5. FSAW METHOD BY PREFERENCE RATIO Assume there are m alternatives and n attributes and let xij represents the value of ith alternative measured by jth attribute. Then, a matrix of (m.n) represents the system. We also assume wj is the importance weight of jth attribute, for j= 1,…,n. However, it is clear that neither xij nor wj is necessarily a crisp number and usually are expressed by a fuzzy number, including in linguistic terms. In this case, assigning a score to each alternative is not straight forward any more. To see that, let xij and wj be defined as follows.

w j = { y j , µ wj ( y i )} j = 1,..., m. And,

xij = {(rij , µ xij (rij ))}, i = 1,..., n, j = 1..., m. Then, if the utility of alternative Ai is represented by Si, then,

Si =

∑ nj =i y j r ij ∑ nj =1 y j

In this section by summing up the previous algorithms we suggest a method for FSAW. The final aim is to assign a crisp score of Si to alternative i, for i = 1, 2, …, n, in order to rank them. Step 1. If the importance weight of jth attribute is represented by a fuzzy number wj, for j = 1, 2, …, n, then normalize them by applying algorithm III. Let wj be the normalized importance weight of jth attribute in crisp number. xij is the value of alternative i measured by Step 2. Evaluate each alternative by all attributes. If ~

attribute j and expressed as a fuzzy number, then compare ~ xij , j = 1, 2, …, m, by steps 1 and 2 of algorithm

III. Let xij represent the value of alternative i measured by attribute j after that. Step 3. Assign a crisp score as follows, to alternative i.

S i =∑ j =1n wi xij Example 3 Consider the example introduced by Bonissone[2]. There are three alternatives and four attributes. The importance weighs for attributes are expressed in terms of trapezoidal fuzzy numbers, as follows.

~ W = [(0.6, 0.8, 0.8, 1), (0.6, 0.8, 0.8, 1), (0.8, 1, 1, 1), (0, 0.2, 0.2, 0.4)]. Table II shows the value of four alternatives measured by three attributes. Each element of the matrix is a trapezoidal fuzzy numbers. Applying the method suggested by Bonissone[2] results in the following fuzzy scores U1 = (0.64, 1.26, 1.34, 2), U2 = (0.6, 1.46, 1.46, 2.26) and U3 = (1.32, 2.32, 2.42, 2.94). Therefore, using Bonissone's method does not rank the alternatives directly and need another fuzzy ranking method to rank these three numbers. Table II. The matrix of xij for example 3. Alternative A1 A2 A3

Objective 1 (0, 0, 0.1, 0.3) (0.3, 0.5, 0.5, 0.7) (0.7, 0.9, 1, 1)

Objective 2 (0, 0.2, 0.2, 0.4) (0.3, 0.5, 0.5, 0.7) (0.8, 1, 1, 1)

Objective 3 (0.8, 1, 1, 1) (0.3, 0.5, 0.5, 0.7) (0.6, 0.8, 0.8, 1)

Now we apply the proposed method, which results in crisp scores directly.

Objective 4 (0.3, 0.5, 0.5, 0.7) (0.6, 0.8, 0.8, 1) (0, 0, 0.1, 0.3)

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243

~

~

Step 1. By applying algorithm III, the weights are normalized. First A is defined as A = (0.8, 1, 1, 1). Then, k = (k1, k2, k3, k4)= (0.85, 0.85, 1, 0.213) Finally, W = (0.2918, 0.2918, 0.3433, 0.073). Step 2. By applying algorithm III, the attributes are compared. Alternative A1 A2 A3

Objective 1 0.1342 0.5536 1

Objective 2 0.213 0.532 1

Objective 3 1 0.532 0.85

Objective 4 0.622 1 0.1753

Step 3. The score of each alternative is obtained by (3). Alternative A1 A2 A3

Score 0.49 0.572 0.888

Rank 3 2 1

As can be seen the scores are within [0,1] range and can be compared with "1", as the bench mark. In Bonissone's method the scores do not lie necessarily in that interval.

6. CONCLUSION In this paper we developed a new method for fuzzy multiple attribute decision making (FMADM) within the framework of simple additive weighting method. This method is developed by applying the concept of preference ratio, thus it is easy as well as realistic. We also proposed three other algorithms for ranking and normalizing fuzzy numbers. The concept of preference ratio can also be applied to develop other methods for fuzzy multiple attribute decision making (FMADM) such as fuzzy TOPSIS or fuzzy AHP.

ACKNOWLEDGEMENT The authors would like to thank the anonymous reviewers for their constructive suggestions.

REFERENCE 1.

S. M. Bass and H. Kwakernaak. Rating and ranking of multiple aspect alternative using fuzzy sets, Automatica, 13, 47-58, 1977. 2. P. P. Bonissone. A pattern recognition approach to the problem of linguistic approximation in system analysis, IEEE, International conference on cybernetics and society, 793-798, 1979. 3. S. J. Chen and C. L. Hwang. Fuzzy Multiple Attribute Decision Making, Methods and Applications, (Springer-Verlag), 1992. 4. Y. M. Cheng and B. McInnis. An algorithm for multiple attribute decision problem based on fuzzy sets with application to medical diagnosis, IEEE transaction on System, Man and Cybernetics, SCM10, 645-650, 1980. 5. W. M. Dong and F.S. Wong. Fuzzy weighted average and implementation of the extension principle, Fuzzy Sets and Systems, 21, 183-199, 1987. 6. D. Dubois and H. Prade. Ranking of fuzzy numbers in the setting of possibility theory, Information Sciences, 30, 183-224, 1983. 7. D. Dubois and H. Prade. Fundamentals of Fuzzy Sets, Kluwer Academic Publishers, Boston, 2001. 8. C.L. Hwang and K. Yoon. Multiple Attribute Decision Making-Methods and applications, A state-ofthe-arts survey, Springer- Verlag, New York, 1987. 9. H. Kwakernaak. An Algorithm for rating muliple-aspect alternatives using fuzzy sets. Automatica, 15, 615-616, 1979. 10. M. Modarres and S. Sadi-Nezhad. Ranking fuzzy numbers by preference ratio, Fuzzy Sets and Systems 118, 429-436, 2001.

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11. Tian-Shyliou and M.J. Wang. Fuzzy weighted average, An improved algorithm, Fuzzy Sets and Systems, 49, 307-315, 1992. 12. Y. Yuan. Criteria for evaluating fuzzy ranking methods, Fuzzy Sets and Systems, 44, 139-157, 1991. 13. Lai and C.L. Hwang, Fuzzy Mathematical Programming, Methods and Applications, Springer-Verlag, 1992. 14. T.Y. Tseng and C.M. Klein. Algorithm for ranking procedure in fuzzy decision making, IEEE transaction on System, Man and Cybernetics, 19, 1289-1296, 1989.

ABOUT THE AUTHOR(S) M. MODARRES is Professor at Sharif University of Technology, Department of Industrial Engineering, in Tehran, Iran. He has a Ph.D, from the University of California, Los Angeles, Operations Research, 1975, M.S, University of California, Engineering System, 1973, M.S, Tehran University, Electrical Engineering, 1968. His Publications include European Journal of Operational Research, IEEE Transactions on Power Systems, Naval Research Logistics Quarterly, Fuzzy sets and systems, Journal of Operational Research Society, Applied Mathematics and Computation, International Journal of Uncertainty, Fuzziness and Journal of Engineering, Systems, Journal of Engineering, Iranian Journal of Science and Technology, International Journal of Inquiry, Scientia Iranica. His research interests are Stochastic Models, Revenue Management, Supply Chain Management, and Fuzzy Numbers Models.

S. Sadi-Nezhad is Head Of Education Department, MBA Faculty, Industrial Management Institute, Tehran, Iran. He has a PhD, MS, and BS in Industrial Engineering (1999, 1989, and 1987) Tehran, IRAN. His publications include journals such as Fuzzy sets and systems; ModirSaz (Iran); Third, Fourth & Fifth International Conference on Fuzzy ICAFS. His research interests include Fuzzy Multiple Criteria Decision Making, Systems Thinking, Fuzzy Ranking, Fuzzy Production Planning & Inventory Control, and Fuzzy Scheduling