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A decision-making method based on interval-valued fuzzy sets for cloud service evaluation

A decision-making method based on interval-valued

fuzzy sets for cloud service evaluation

Chen-Tung Chen Department of Information Management National United University

MiaoLi, Taiwan

ctchen@https://www.wendangku.net/doc/ca14299303.html,.tw

Kuan-Hung Lin

Master Program in Business Administration National United University

MiaoLi, Taiwan

KuanHungLin@https://www.wendangku.net/doc/ca14299303.html,

Abstract—In the real world, many decision-making problems contain uncertain data in the decision process. Although fuzzy set theory can deal with the uncertain problem, the decision maker always cannot express his opinion exactly in the decision-making process. Because the subjective judgments of decision makers are always vagueness, linguistic variables and interval-valued fuzzy sets are suitable used to represent the fuzziness of membership value of their opinions. In this paper, the fuzzy AHP method is applied to compute the fuzzy weights of each criterion based on the interval-valued fuzzy sets. The 2-tuple linguistic variables are used to express the performance rating of all alternatives. and then, determine the ranking order of all alternatives in accordance with preference matrix. This proposed method can provide decision maker more flexibility to express his subjective judgment with respect to each alternative. Finally, an example is presented to show the procedure of the proposed method at the end of the paper.

Keywords-Fuzzy AHP; Interval-valued fuzzy sert; cloud computing;

I.I NTRODUCTION

Decision making is one of the most complex administrative processes in management [1]. In some situations, because of time pressure, lack of knowledge, and the decision maker’s limited attention and information processing capabilities [20], the decision maker cannot make the optimum decision. In the decision process, we may both consider several qualitative and quantitative criteria simultaneously, this may make the selection process more complex and full challenging.

In order to solve this problem, Saaty [15] proposed analytic hierarchy process (AHP) method for solving complex problem to hierarchical structure and systematization. In AHP, an expert is asked to give ratios for each pairwise comparison between issues for each criterion in a hierarchy, and also between the criteria [3]. The pairwise comparison result display in a hierarchy with a weight for each criterion, provide both qualitative as well as quantitative characteristics here.

Although AHP have many advantage for decision making, it still has some disadvantages such as the subjective judgments of the decision makers are often uncertain in determining the pairwise comparison. Buckley [2][3] proposed the fuzzy AHP to overcome the disadvantages. This method allowed the experts to use fuzzy ratios in place of exact ratios.

However, it is often difficult for an expert to exactly quantify his or her opinion as a number in interval [0, 1]. Therefore, it is more suitable to represent this degree of certainty by an interval [1][18]. The membership value express in interval value can more fit the real world situations. Therefore, interval-valued fuzzy sets (IVFs) are suitable used to represent the fuzziness of membership value of their opinions. We combined IVFs with fuzzy AHP to propose a new method to determine the fuzzy weights in interval-valued fuzzy numbers.

The 2-tuple linguistic representation model is based on the concept of symbolic translation [9][19], decision makers can apply 2-tuple linguistic variables to evaluate the performance level of alternative in criteria, it is an effective method to reduce the mistakes of information translation and avoid information loss through computing with words [10].

In this paper, we proposed a new decision method based on using IVFs combined fuzzy AHP to determine the weights of criteria, and using 2-tuple linguistic variables determine the performance of alternative in criteria. This can provides decision maker have more flexibility to evaluate decision alternative.

II.F UZZY S ET T HEORY

A fuzzy set can be defined mathematically by assigning to each possible element in the universe of discourse a value representing its grade of membership in the fuzzy set [4][22]. The fuzzy number A

is a fuzzy set whose membership function )

(~x

satisfies the following conditions [4][13]:

(1). )

(~x

is piecewise continuous;

(2). )

(~x

is a convex fuzzy subset;

(3). )

(~x

is normality of a fuzzy subset.

Triangular fuzzy number (TFN) can be defined as)

=, where u

<.When l > 0, then T

is a positive TFN (PTFN) [6][23]. The membership function of positive TFN T

is defined as follow (show in Fig. 1).

?????????≤≤??≤≤??=otherwise u x m m u x u m x l l

m l

x x T ,0,,)(~μ (1)

Figure 1. Positive TFN.

The α-cut is used to transform fuzzy set into crisp interval

[23]. The α-cut is defined as follow.

10],)(,)[(~≤≤??+?=αααα

m u u l l m T (2) In short, we can represent ],[~α

ααu l T T T = as Fig. 2.

Figure 2. The α-cut for positive TFN.

Give two PTFNs ),,(~1111u m l T = and ),,(~2222u m l T =.

The additive (⊕) and multiply (?) operations between them

can be expressed as follows [12].

),,(~

~21212121u u m m l l T T +++=⊕ (3) ),,(~~21212121u u m m l l T T ×××=? (4)

Lee and Li [14] proposed the generalized mean value method to defuzzy the fuzzy number. Let ),,~u m l T (=be a

TFN, the defuzzied formula can be shown as follow. 3/)()~(u m l T G ++= (5)

18/])([)~(2

22mn nl lm n m l T S ???++= (6) where )~(T G is the generalized mean value and )~

(T S is the deviation of fuzzy number of T ~, respectively. IVFs were suggested for the first time by Gorzlczany [7] and Turksen [17]. Because of the expert can not exactly give their opinion in interval [0, 1], use the interval to represent this uncertain is suitable. In order to express the fuzziness more exactly, [1] use interval-valued triangular fuzzy number

(IVTFN) to present the opinions of decision-makers. It allows membership value represent in lower limit of degree of membership L A μ and upper limit of degree of membership U

A μ.

In some case, an expert cannot exactly express membership value; the IVTFN can express the uncertain in a range (as the Fig. 3). The original upper limit of membership value is u , it becomes to [u ,uu ] to represent the fuzziness now. The IVFs

defined as follow [1]: )])}(),([,{(~x x x T U T L T μμ= (7)

T L T U T L T X x X μμμμ<∈?→,],1,0[:, (8)

Figure 3. Interval-valued triangular fuzzy number. A 2-tuple linguistic variable can be expressed as ),(i i s α, where i s represent the i -th linguistic term in a finite and totally

ordered linguistic term set },...,,,{210g s s s s S =, and i α is a

numerical value which is the difference between calculated

linguistic term and i s [9]. The type of linguistic variable set is provided in Table I. Their membership functions are shown as Fig. 4.

Figure 4. Membership functions of linguistic variables. The symbolic translation process is applied to translate β (β∈ [0, 1]) into a 2-tuple linguistic variable. The generalized translation function (Δ) can be computed as [16]: )21,21[]1,0[:g g S ?×→Δ (9)

?????

?∈?=?==Δ)21,21[,,),()(g g g i g) round(i s with s i i αβαβαβ (10) where Δ is the symbol represented β translated into a 2-tuple linguistic variable.

A reverse equation 1?Δ is necessary to return an equivalent

numerical value, a 2-tuple linguistic variable translates into crisp value β (β∈ [0, 1]), computed as [16]:

i i i g i

s αβα+==Δ?),(1 (11) where 1?Δ is defined to return an equivalent numerical value β from 2-tuple linguistic information ),(i

i s α.

(70s )

1 13/6 (73s ) (7

6s ) 2/65/6

(72s ) (74s ) (71s ) (75

s ) 1/64/6 01U T ~

μll uu

m u 0

)(~x T μL T

~μ1

m u

l T

~u T

Let (s 1, α1), …, (s n , αn ) be a 2-tuple linguistic variable set, the arithmetic mean is computed as [10]:

)),(1(),(1

∑ΔΔ==?n i i i s n s αα (12)

III. P ROPOSED M ETHOD

Buckley [2][3] proposed the fuzzy analytic hierarchy

process method to compute the fuzzy weights in MCDM problems. It is a systematic approach to the alternative selection and justification problem by using fuzzy set theory and hierarchical structure analysis [8]. It allows experts specify importance between each criterion and alternative in a hierarchy framework, and acquire weights for each criterion. The fuzzy AHP method was popular in solving MADM problem, but still has some problems. For example, the expert in determining the membership value of criteria or alternative, they may not express exactly, the membership value might be an interval. Therefore, expert can express his judgment more flexibility by using the IVFs.

Based on IVFs, we applied Buckley’s method to present a new way to compute the fuzzy weight of each criterion. And then, using 2-tuple linguistic variable to evaluating alternative. We applied these two methods, present a new way to solve the decision problem. The computation procedure is shown as follows.

Step 1. Establishing hierarchical structure.

Establish a hierarchy framework of decision-making problem based on the objective, literature review and experts’ opinions. Within the hierarchy framework, there may contains several criterion, sub-criterion and alternatives.

Step 2. Experts judgment from the hierarchical structure. The experts pairwise comparison in the criteria and sub-criteria, and judgment the alternatives performance in every sub-criterion.

Step 3. Using interval-valued fuzzy AHP obtain fuzzy weights.

Using interval-valued fuzzy AHP, we can calculate the criteria and sub-criteria weights which present in IVTFN. The procedure of calculation steps as follow.

A. Constructing the fuzzy positive reciprocal matrix. The experts are allowed to use interval-valued fuzzy ratios (see

Table II) in place of exact ratios. The ij T ~

, i ≠ j, can now be

IVTFN in any positive reciprocal matrix. The fuzzy positive reciprocal matrix is defined as

n n ij T T ×=~

~ (13)

j i T ij =?=,1~

(14) n j i T T ij

ji ,...,2,1,,~1~

=?= (15) If ]),[,],([~

uu u m l ll T ij =, then ]),[,],,([~111111??????=ll l m u uu T ij

B. Computing the interval-valued fuzzy weights of criteria.

According to the Lambda-Max method [5], a sequence of fuzzy

positive reciprocal matrix is applied to determine the relative weights of each criterion. The procedure of all calculations steps as follow.

(1) Let α = 1, using α-cut to obtain []

n n ijm m t T ×=1~, it represent the decision-makers give the crisp positive reciprocal

matrix T ~, and determine the fuzzy weight m W ~

, for []n i W W im m ,...,2,1,~

==.

(2) Let α = 0, using α-cut to obtain the []n

n iju

u t T ×=0~

, []

n n ijuu uu t T ×=0~, []

n n ijll ll t T ×=0~

, and []n

n ijl

l t T ×=0 in accordance with the fuzzy positive reciprocal matrix. Next,

computing the fuzzy weights ll W ~,l W ~ , u W ~ and uu W ~

, where []ill ll W W =~, []il l W W =~, []iu u W W =~, and []n i W W iuu uu ,...,2,1,~

==.

(3) In order to make sure the fuzzy weights are still normal fuzzy sets, uses the following formula adjust these weights. According to the method of [5], first, we should determine lower bounds triangular fuzzy weights. }1|min{n i W W

Q il

im l ≤≤= (16)

}1|max{n i W W

Q iu

im u ≤≤= (17)

When find the constants l Q and u Q , to calculate the new weight as follow. il l il W Q W ×=* (18)

iu u iu W Q W ×=* (19)

Second, we use the new fuzzy weight of lower limit and upper limit to determine the upper bounds triangular fuzzy weights as follow.

}1|min{*n i W W Q ill

ll ≤≤= (20)

}1|max{*n i W W Q iuu

uu ≤≤= (21)

Then, using the constants ll Q and uu Q

to determine the new upper bounds triangular fuzzy weights as follow.

ill ll ill W Q W ×=* (22)

iuu uu iuu W Q W ×=* (23) Finally, we can get the new fuzzy weights ][*

*ill ll W W =, ][**il l W W =, ][**iu u W W =, and ][*

*iuu uu W W =, for i=1,2,…n.

(4) To integrate *ll W , *l W , m W , *u W and *

uu W , we can obtain the positive triangular fuzzy weight matrix for decision-makers. The fuzzy weights of i -th criterion can be represented

as ]),[,],,([~****iuu iu im il ill i W W W W W W =. (5) Defuzzy the interval-valued fuzzy weights

In this paper, we applied the method of [14] to transform the interval-valued fuzzy weight into interval weight. If the fuzzy weight of criterion j is ]),[,],,([~

juu ju jm jl jll j W W W W W W =, then transform the

interval-value fuzzy weight into crisp interval value as ],[max min j j j W W W = (24) where

/)(min ju jm jll j W W W W ++= and

3/)(max juu jm jl j W W W W ++=. C. Computing the interval-valued fuzzy weights of sub-criteria. According to the fuzzy positive reciprocal matrix for each sub-criteria pairwise comparison in each criterion, repeating the process of step B to obtain the sub-criteria

weights of each criterion.

Step 4. Computing the alternative performance rating.

According to p -th expert give to the performance ratings

ijk

X of alternative A i with respect to sub-criteria jk C , the aggregated linguistic rating (ijk

x ~) of alternatives can be

calculated by using arithmetic mean as

),()),(1(~1

1ijk ijk t ijk P t t ijk ijk S S P x αα=∑ΔΔ==? (25)

where i =1, 2, …, m ; j =1, 2, …, J ; k =1, 2, …, K j . Then, we can translate aggregated linguistic ratings into crisp value β to represent the performance rating of the alternative. Step 5. Integrate criteria weights and alternative performance rating.

According to above steps, we can obtain the criteria weights and alternative performance rating. The overall score

of alternatives can be calculated as follow. A. Multiply the sub-criteria weights and performance rating of each alternative. Let the sub-criteria weights multiply

alternative performance rating, we can get the j -th criterion score, the calculation as follow. ∑∑×Δ===?j

k jku jkl j K k jku jkl ijk ijk ij W W W W S 1

],[],[),(α (26) where ij X represents the score of i -th alternative with respect

to j -th criterion. The ],[jku jkl W W represents the interval fuzzy weight of k -th sub-criterion under j -th criterion.

B. Multiply the criteria weights and criteria score. Let the

criteria weights multiply criteria score, we can get the score of i-th alternative for overall objective, the calculation as follow. ∑∑×=

==J

j j j J j j j ij i W W W W X S 1

max min 1

max min ]

,[]

,[ (27) Step 6. Ranking alternative.

According to the above-mentioned processes, we can get the final score of each alternative. Let the final score of A i be

],[+?=i i i S S S , final score of A j be ],[+?=j j j S S S . Using ranking method of [21] to compare two interval values, the

degree of possibility that alternative i A superior to j A

()(j i A A P ≥) can be computed as ??

???????????????????+???=≥?+?+?+0,0,max 1max )(j j i i i

j j i S S S S S S A A P (28)

According to the degree of possibility, we applying Hsu

and Chen [11] ranking procedure to ranking alternatives. The procedure is shown as follow. A. Constructing relation matrix According to the degree of

possibility that alternative i A superior to j A , we can

Construct the relation matrix as follow.

m m ij e E ×=][ (29) where )(j i ij A A P e ≥=.

B. Calculate the strict relation matrix. The value of s

e indicates the degree o

f strict dominance of alternative i A superior to j A . m m s

ij s e E ×=][ (30)

where ?

??≥?=otherwise e e when e e e ji ij ji ij s

ij ,0, C. Calculate the nondominated degree of each alternative. The nondominated degree of each alternative i A (i =1,…, m )

can be defined as s ji i j j s ji i j j i ND e e A ≠Ω

∈≠Ω∈?=?=max 1}1{min )(μ (31)

where )

(i ND A μ indicates i A has the highest nondominated degree in all alternative Ω.

D. Determine the alternatives preference. We can use the

)(i ND A μ value to determine the preference, the procedure is described as follows: (1) Set K = 0 and Ω={A 1, A 2, …, A m }. (2) Select the alternatives (h A ) which have the highest

nondominated degree, where )}({max )(i ND

i h ND A A μμ=

and the set the preference of h A is 1)(+=K A r h .

(3) Delete the alternatives form Ω and the corresponding row and column of h A deleted from the strict relation matrix. (4) Recalculate the nondominated degree for each

alternative i A , while ?=Ω, then stop; otherwise, set K =K + 1 and return to step (2).

IV.N UMERICAL E XAMPLE

Cloud computing combined with internet architecture and high-speed calculating ability to come with the tide of fashion. Due to cloud computing characteristics of economics, simplification and convenience, many enterprises abandon tradition information services model gradually and transform into cloud computing services model.

Suppose that a company desires to select a cloud computing service to serve their customer, After preliminary screening, three cloud computing supplier A1, A2, A3 remain for further evaluation. A committee of three decision-makers (or experts) (p1, p2, p3) has been formed to select the most suitable supplier.

In this example, 9 sub-criteria (C11, C12, …, C33) are considered to select the suitable supplier. The computation process of proposed method is shown as follows.

Step 1. According to description of the problem, the experts construct hierarchical structure of this problem is constructed as Fig.5.

Figure 5. The evaluation model.

Step 2. Each expert uses the linguistic variables to pairwise comparison and evaluate weights of each criterion and sub-criterion; Using 2-tuple linguistic variable to evaluate each alternative of each sub-criterion. The original values shown as Table III and Table IV.

Step 3. Computing interval-valued fuzzy weights.

A. The expert using linguistic variables to constructing the fuzzy positive reciprocal matrix shown as Table III.

B. The weights of criteria can be computed as W1=[0.422,

0.454], W 2=[0.256, 0.264] and W3=[0.287, 0.303].

C. The sub-criteria weights shown as Table V.

Step 4. The alternative aggregated linguistic ratings and crisp value β shown as Table VI.

Step 5. Integrate criteria weights and alternative performance rating as follow.

A. Multiply the sub-criteria weights and alternative performance rating, the result show as Table VII.

B. The final scores of these alternatives can be calculated as A1=[0.426, 0.549], A2=[0.461, 0.597] and A3=[0.408, 0.528].

Step 6. Ranking alternative.

A. According to the degree of possibility, the relation matrix is constructed as Table VIII.

B. Calculate the strict relation matrix, that i A strict superior to j A, the result shown as Table IX.

C. Calculate the nondominated degree of each alternative, in Table IX as 676

)

(1=

μ , 0.1

)

(2=

μ and

524

)

(3=

μ.

D. After the ranking procedure, the final preference is

A f

f. According to the result, decision-maker may consider to choice the alternative A2.

V.C ONCLUSIONS

In general, decision-making process will encounter many uncertain situations. Although applying fuzzy AHP, we can evaluate the problem into hierarchical structure and systematization; however, decision makers always difficult to express their opinion into membership value exactly. Under this situation, the IVFs is suitable to express the subjective opinion of each expert.

In this paper, we applied the fuzzy AHP method to compute the fuzzy weights of all criteria and using 2-tuple linguistic variable evaluate alternative. Applying this method can let decision maker using linguistic variable easily to determine the criteria importance and alternative performance rating, and systematic solve decision problem.

In the future, we will design a decision analysis system based on this proposed method to reduce the computation time and improve the decision-making quality.

A CKNOWLEDGMENT

The authors gratefully acknowledge the financial support partly by the National Science Council (NSC), Taiwan, under project number NSC-97-2410-H-239-009-MY2.

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TABLE I.

THE 2-TUPLE LINGUISTIC VARIABLES

Linguistic variable

Extremely Poor )(70s , Very Poor )(71s , Poor )(7

2s , Fair )(73

s , Good )(7

4s , Very Good )(75

s , Extremely Good )(76s TABLE II.

LINGUISTIC VARIABLE FOR IMPORTANCE Symbol

Linguistic variables fuzzy number c Extremely Equal Importance

([1,1] ,1, [1,1])d Equal Importance ([1,1] ,1, [2,3])e Intermediate values ([1,1] ,2, [3,4])f Weak Importance ([1,2] ,3, [4,5])g Intermediate values ([2,3] ,4, [5,6])h Essential Importance ([3,4] ,5, [6,7])i Intermediate values ([4,5] ,6, [7,8])j Very Strong Importance ([5,6] ,7, [8,9])k Intermediate values ([6,7] ,8, [9,9])l

Absolute Importance

([7,8] ,9, [9,9])

TABLE III.

ORIGINAL LINGUISTIC VARIABLE FOR IMPORTANCE

PAIRWISE COMPARISON

C 1 C 2 C 3 C 11 C 12 C 13 C 21 C 22 C 23 C 31C 32C 33C 1 c e g C 11 c h -1 j -1 C 21c f h -1 C 31 c g -1e C 2 e -1 c d -1 C 12 h c e -1 C 22f -1 c l -1 C 32 g c k P 1

C 3 g -1 d c C 13 j e c C 23h l c C 33 e -1k -1c C 1 C 2 C 3 C 11 C 12 C 13 C 21 C 22 C 23 C 31C 32C 33C 1

c h g C 11 c e -1

d -1 C 21c i -1 f C 31 c g

e C 2 h -1 c e C 12 e c g -1 C 22i c j C 32 g -1c j P 2 C 3 g -1 e -1 c C 13 d g c C 23

f -1 j -1 c C 33 e -1j -1c C 1 C 2 C 3 C 11 C 12 C 13 C 21 C 22 C 23 C 31C 32C 33C 1

c i -1 h -1 C 11 c f e C 21c g k C 31 c e -1k -1C 2 i c e -1 C 12 f -1 c g C 22g -1 c e C 32 e c g -1P 3 C 3 h

C 13

e -1

g -1

C 23

k -1

e -1

C 33

TABLE IV.

THE RATING LINGUISTIC VARIABLE WHICH ARE

EVALUATED BY EXPERTS

P 1 P 2 P 3

A 1 A 2 A 3 A 1 A 2 A 3 A 1 A 2 A 3 C 11VG F F EG P VP EG P F C 12P EP P VG P P P VP VG C 13 F G VP P F F F P VP C 21VP VG VG VP P P P VG G C 22EP F F P G VP VP F F C 23G VP P F G VG VG VG F C 31EG P G P EG EG EG VG EG C 32EP VG EG EP VG F F G VP C 33

EG EG EP VP P P P P P TABLE V.

THE SUB-CRITERIA WEIGHT

Sub-

criteria

Weights

Sub- criteria Weights Sub-

criteria Weights

C 11 [0.261,0.283]C 21 [0.347,0.366] C 31 [0.267,0.285]C 12 [0.299,0.316]C 22 [0.259,0.341] C 32 [0.409,0.424]C 13 [0.416,0.439]

C 23 [0.289,0.307] C 33 [0.308,0.317]

TABLE VI.

THE AGGREGATED LINGUISTIC RATINGS

A 1

A 2

A 3

C11)06.0,(76?s 0.944)06.0,(72s 0.389 )06.0,(72s 0.389

C12

)0,(73s

0.500

)0,(7

1s 0.167

)0,(7

0.500

C13)06.0,(7

3?s 0.444)0,(73s 0.500 )06.0,(72?s 0.278C21)06.0,(71s 0.222

)0,(74s

0.667 )06.0,(74?s 0.611

C22)0,(7

1s 0.167)06.0,(73s 0.556 )06.0,(72s 0.389C23

)0,(74s

0.667)06.0,(73s 0.556 )06.0,(7

3s 0.556

C31)06.0,(75?s 0.778)06.0,(74s 0.722 )06.0,(75s 0.889

C32)0,(7

1s 0.167)06.0,(75?s 0.778 )06.0,(7

3s 0.556C33

)0,(73s

0.500)06.0,(73s 0.556 )06.0,(71s 0.222

TABLE VII. THE ALTERNATIVE SCORE IN EACH CRITERION

Criterion A 1 A 2 A 3

C 1 [0.558,0.638][0.345,0.393] [0.352,0.401]C 2 [0.309,0.383][0.529,0.675] [0.467,0.588]C 3 [0.419,0.459][0.664,0.724] [0.519,0.569]

TABLE VIII. THE REALATION MATRIX

A 1A 2 A 3 A 10.500 0.338 0.578 A 20.662 0.500 0.738 A 30.422 0.262 0.500

TABLE IX.

THE STRICT REALATION MATRIX

A 1 A 2 A 3 A 10 0 0.156 A 20.3240 0.476 A 3

0 0 0