Group decision-making using a fuzzy linguistic ... - Semantic Scholar
European Journal of Operational Research 154 (2004) 563–572 www.elsevier.com/locate/dsw
Decision Aiding
Group decision-making using a fuzzy linguistic approach for evaluating the flexibility in a manufacturing system Reay-Chen Wang *, Shian-Jong Chuu Department of Industrial Management, National Taiwan University of Science and Technology, 43, Keelung Road, Section 4, Taipei (106), Taiwan, ROC Received 30 July 2001; accepted 18 September 2002
Abstract This study is to build a group decision-making structure model of flexibility in a manufacturing system development. The paper also proposes two algorithms for determining the degree of manufacturing flexibility (MF) in a fuzzy environment using a fuzzy linguistic approach. While evaluating the degree of MF, one may find the need for improving MF, and determine the dimensions of MF as the best direction to improvement until she/he can accept it The results of this study are more objective and unbiased since they are generated by a group of evaluators. Ó 2002 Elsevier B.V. All rights reserved. Keywords: Manufacturing flexibility; LOWA operators; Group decision-making; Linguistic fuzzy quantifier; Maximum entropy weights
1. Introduction Manufacturing environments have changed so fast in recent decades that the strategic function of manufacturing systems have become increasingly important. Flexible manufacturing, computerintegrate manufacturing and Just-in-Time systems depend on flexibility [15]. Flexibility is emerging as a key competitive priority in todayÕs manufacturing systems.
*
Corresponding author. Tel.: +886-2-27376353; fax: +886-227376344. E-mail address: [email protected] (R.-C. Wang).
Generally, manufacturing flexibility (MF) is the ability of a manufacturing system to cope with environmental changes effectively and efficiently. Operation managers must evaluate MF when making capital investment decisions and measuring performance level [6]. MF is a complex, multidimensional and difficult-to-synthesize concept [17], and so the needs of operation managers have not yet been met. Many reviews have considered definitions of MF, requests for MF, classificatory dimensions of MF, measurement of MF, choices for MF, and interpretations of MF [1,3,8,16,17,21]. GerwinÕs [6] conceptual framework concerns the influence of environmental uncertainty on MF. Upton [20] presented a framework for analyzing MF using
0377-2217/$ - see front matter Ó 2002 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(02)00729-4
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different dimensions, each of which at different time intervals and is specified by three elements–– range, mobility and uniformity. Golden and Powell [7] proposed an inclusive definition in which flexibility can be measured by four metrics––efficiency, responsiveness, versatility and robustness. Many researchers have tried objectively to quantify MF. Several efforts are theoretical and involve only two or three dimensions of MF [1]. The importance of dimensions of flexibility and the subjective evaluation of MF have seldom been addressed. Most operation managers cannot provide exact numerical values to express opinions based on human perception: more realistic measurement uses linguistic assessments instead of numerical values [1,6,10,13,21]. In fact metrics can be measured as linguistic labels (terms) such as very high, high, middle, low, and very low. After Zadeh [23] introduced fuzzy set theory to deal with vague problems, linguistic labels have been used in approximate reasoning within the framework of fuzzy set theory [24] to handle the ambiguity in evaluating data and the vagueness of linguistic expression. Normal trapezoidal fuzzy numbers have been used to characterize linguistic terms used in approximate reasoning. Therefore, the purpose of this study was to build a group decision-making structure model of flexibility in a manufacturing system. Two algorithms are proposed to assess the degree of MF in a fuzzy environment using a fuzzy linguistic approach to any phase of the life cycle. Section 2 presents a fuzzy linguistic approach to evaluating flexibility in a manufacturing system development. Section 3 proposes a hierarchical structure model of flexibility in a manufacturing system development. Assume that a group of n decision makers (D1 ; D2 ; . . . ; Dn ) are responsible for assessing the degree of MF for the management. One method aggregates each parameter assessed by an individual, and aggregates the results to determine the final degree of flexibility. Another method aggregates the degrees of flexibility determined by individuals and then aggregates them as a final degree. Sections 4 and 5 separately consider these two algorithms. Finally, Section 6 presents the numerical examples.
2. Fuzzy linguistic approach 2.1. Linguistic assessments The fuzzy linguistic approach assesses linguistic variables using words or sentences of a natural language [24]. This approach is appropriate for some problems in which information may be qualitative, or quantitative information may not be stated precisely, since either it is unavailable or the cost of its determination is excessive, such that an Ôapproximate valueÕ suffices [10]. Most experts cannot give exact numerical values to express their opinions on flexibility metrics. More appropriately, measurements are stated as linguistic assessments rather than numerical values. When applying a fuzzy linguistic approach to measuring MF, only the importance of flexibility dimensions, based on the manufacturing strategy [6] is considered. The flexibility metrics are considered to be equally important, since certain measures of flexibility are still not widely accepted [1,3,7,21]. For example, when determining MF, Ômix flexibilityÕ is frequently an important factor. Therefore, performance should be measured according to each flexibility metric, and the importance of each flexibility dimension should be determined. As mentioned above, the rating of performance and grade of importance should be rated for each item. Therefore, both were scored on a nine-rank scale, as shown in Table 1. Let S ¼ fs0 ; s1 ; . . . ; s8 g be a finite and totally ordered term set on ½0; 1 with an odd cardinal, where the middle label, s4 , represents ÔaverageÕ, and the remaining terms are placed symmetrically around s4 , and exhibit the following properties [12]. 1. The set is ordered: si = sj if i = j. 2. The negation operator is defined as Negðsi Þ ¼ sj such that j ¼ 8 i. 3. The maximization operator is Maxðsi ; sj Þ ¼ si if si = sj . 4. The minimization operator is Minðsi ; sj Þ ¼ si if si 5 sj . The nine linguistic labels in S ¼ fs0 ; s1 ; . . . ; s8 g were specified. The semantics associated with each
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Table 1 Linguistic labels of rating of performance and grade of importance Nine ranks of rating of performance S0 S1 S2 S3 S4 S5 S6 S7 S8
Nine ranks of grade of importance
¼ DL: definitely low ¼ VL: very low ¼ L: low ¼ ML: more or less low ¼ M: middle ¼ MH: more or less high ¼ H : high ¼ VH: very high ¼ DH: definitely high
S0 S1 S2 S3 S4 S5 S6 S7 S8
label si are given as a linear trapezoidal membership function associated with a normal fuzzy number. Not all individuals agree on the same associations between membership functions and linguistic labels. However, this paper consider a situation in which experts can perfectly distinguish among the set of labels under a similar conception, and can use linguistic labels to express their opinions. 2.2. Combining linguistic labels This work applies a linguistic ordered weighted averaging (LOWA) operator, which uses the maximum entropy weights, of linguistic labels by direct computation on labels; that is, the independently of the semantics of the label set. The LOWA operator defined by Herrera et al. [11], is based on the ordered weighted averaging operator defined by Yager [22], which used in the problem of multiple attribute decision making, and on the convex combination of linguistic labels defined by Delgado et al. [2]. The LOWA operator on linguistic labels acts directly on a finite and total ordered term set. The main problem associated with finite term sets is
¼ DL: definitely low ¼ VL: very low ¼ L: low ¼ ML: more or less low ¼ M: middle ¼ MH: more or less high ¼ H : high ¼ VH: very high ¼ DH: definitely high
that the impact of small changes in the weights on the weighed aggregation can be large, especially when the ÔroundÕ operation is used. Furthermore, in some cases, the aggregation obtained by applying the LOWA operator will produce ÔNot a NumberÕ. For example, aggregation of the above four labels, fDL; VL; H ; VHg using the LOWA operator, is attempted. Associating Ôas many as possibleÕ, ÔmostÕ and Ôat least halfÕ, with parameters ða; bÞ of ð0:5; 1Þ, ð0:3; 0:8Þ and ð0; 0:5Þ, respectively, yields the results in Table 2. Therefore, the algorithm to determine the maximum entropy weights is applied to reduce the main problem of applying the LOWA operator. Let fa1 ; a2 ; . . . ; am g be a set of linguistic labels to be aggregated. Then, the LOWA operator UQ with the non-decreasing proportional linguistic fuzzy quantifier, Q, is defined as UQ ða1 ; a2 ; . . . ; am Þ ¼ W BT ¼ C m ðwk ; bk ; k ¼ 1; 2; . . . ; mÞ ¼ w1 b1 ð1 w1 Þ C m1 ðbh ; bh ; h ¼ 2; 3; . . . ; mÞ;
ð2:1Þ
where W ¼ ½w1 ; w2 ; . . . ; wm , is a maximum entropy weighting vector, such that, wi 2 ½0; 1 and
Table 2 Comparison between LOWAðWQ Þ and LOWAðW Þ Fuzzy majority quantifiers (Q)
Weighting vector (WQ )
Orness measure (a)
Maximum entropy weighting vector (W )
LOWA (WQ )
LOWA (W )
As many as possible ð0:5; 1Þ Most ð0:3; 0:8Þ At least half ð0; 0:5Þ
f0; 0; 0:05; 0:5g f0; 0:4; 0:5; 0:1g f0:5; 0:5; 0; 0g
0.1667 0.4333 0.8333
f0:0311; 0:0856; 0:2355; 0:6478g f0:1931; 0:2269; 0:2666; 0:3133g f0:6478; 0:2355; 0:0856; 0:0311g
VL ML NaNa
VL ML H
a
Is ÔNot a NumberÕ.
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P wi ¼ 1 (see the Appendix A), bh ¼ wh = h wh ; h ¼ 2; 3; . . . ; m, B ¼ ½b1 ; b2 ; . . . ; bm is the associated ordered label vector, each element bi 2 B is the ith largest label in the ða1 ; a2 ; . . . ; am Þ, C m is the convex combination operator of m labels, is the general product of a label by a positive real number and is the general addition of labels defined in [2]. If m ¼ 2, then C 2 is defined as
P
i
C 2 fwi ; bi ; i ¼ 1; 2g ¼ w1 sj ð1 w1 Þ si ¼ sk ; ð2:2Þ sj ; si 2 S
ðj = iÞ
such that k ¼ Minð8; i þ roundðw1 ðj iÞÞÞ;
ð2:3Þ
where ÔroundÕ is the usual round operation, and b1 ¼ s j , b2 ¼ s i . If wj ¼ 1 and wi ¼ 0 with i 6¼ j for all i, then the convex combination is defined as C m fwi ; bi ;
i ¼ 1; 2; . . . ; mg ¼ bj :
3. Hierarchical structure model of MF A systematic approach is proposed to evaluate the degree of MF, using a fuzzy linguistic approach and hierarchical structure analysis. This method is suited to decision making in a fuzzy environment. The dimensions of flexibility presented by Gerwin [5,6], Slack [18,19], De Toni and Tonchia [3] and Beach et al. [1] were expressed seven dimensions, including mix flexibility, changeover flexibility, and others, according to the relationship between MF and environmental changes. Furthermore, each dimension was divided into four flexibility metrics [7,14], such as efficiency, responsiveness, versatility and robustness. For convenience, the dimension mix flexibility was represented as X1 , changeover flexibility as X2 , and so on. The metrics oh the mth flexibility dimension were represented as in Fig. 1; for example, efficiency was denoted by Xm1 , responsiveness by Xm2 , and so on. The decision makers consider the grade of importance and related rating of performance,
grading both as S ¼ fs0 ; s1 ; . . . ; s8 g. Suppose a group of n decision makers ðD1 ; D2 ; . . . ; Dn Þ are responsible for assessing the degree of MF. The symbol Iðj;mÞ is used to denote the grade of importance of dimension Xm ; Xðj;m;kÞ the rating of performance of flexibility metric Xmk , according to decision maker Dj Õs assessing data (j ¼ 1; 2; . . . ; n; m ¼ 1; 2; . . . ; 7; k ¼ 1; 2; 3; 4). Table 3 represents the above given the data assessed by decision maker Dj (j ¼ 1; 2; . . . ; n). The data assessed by all n decision makers are combined to evaluate the final degree of MF. As in [10], basically two algorithms are considered. A direct algorithm, fD1 ; D2 ; . . . ; Dn g ) solution is based on the individual preference relations. An indirect algorithm, fD1 ; D2 ; . . . ; Dn g ) D ) solution is based on an aggregated preference relation of the group. Therefore, the following two sections of this paper proposes two algorithms for evaluating the degree of MF for use by a group decision makers. 4. Algorithm I Here, an indirect derivation is considered. This algorithm aggregates each parameter assessed by an individual, and aggregates the results to produce the final degree of flexibility. Step I-1: Let Iðj;mÞ and Xðj;m;kÞ be linguistic labels in S ¼ fs0 ; s1 ; . . . ; s8 g. The aggregated parameters obtained from the n decision makersÕ linguistic data can be expressed as follows: IAðmÞ ¼ UQ1 ðIð1;mÞ ; Ið2;mÞ ; . . . ; Iðn;mÞ Þ m ¼ 1; 2; . . . ; 7;
for
XAðm;kÞ ¼ UQ1 ðmð1;m;kÞ ; mð2;m;kÞ ; . . . ; mðn;m;kÞ Þ m ¼ 1; 2; . . . ; 7; k ¼ 1; 2; 3; 4;
ð4:1Þ for ð4:2Þ
where UQ1 is the LOWA operator with the maximum entropy weights, W1 , obtained from the nondecreasing fuzzy linguistic quantifier Q1 , which represents the fuzzy majority over the n decision makers.
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Fig. 1. Hierarchical structure model of MF.
Table 3 The contents of structure model Flexibility dimension (with Q2 ) Grade of importance Flexibility metric (with Q1 ) Rating of performance
X1
X2
X3
X4
X5
X6
X7
IðX1 Þ
IðX2 Þ
IðX3 Þ
IðX4 Þ
IðX5 Þ
IðX6 Þ
IðX7 Þ
X11 X12 X13 X14
X21 X22 X23 X24
X31 X32 X33 X34
X41 X42 X43 X44
X51 X52 X53 X54
X61 X62 X63 X64
X71 X72 X73 X74
m11 m12 m13 m14
m21 m22 m23 m24
m31 m32 m33 m34
m41 m42 m43 m44
m51 m52 m53 m54
m61 m62 m63 m64
m71 m72 m73 m74
Step I-2: First stage assessment. The rating of performance is rated against flexibility metrics using linguistic labels, as in Table 1, which are determined by operation managers. Using the concept of fuzzy majority over the flexibility met-
rics specified by a fuzzy linguistic quantifier Q2 , and using the LOWA operator associated with the maximum entropy weights, W2 , yields the first stage aggregate ratings on dimensions of flexibility, as follows:
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XAðmÞ ¼ UQ2 ðXAðm;1Þ ; XAðm;2Þ ; XAðm;3Þ ; XAðm;4Þ Þ for m ¼ 1; 2; . . . ; 7:
5. Algorithm II ð4:3Þ
Thus, computing the differences between the ranks of the linguistic labels between the grade of importance and the first stage aggregate rating for each flexibility dimension. If XAð2Þ ¼ si and IAð2Þ ¼ sj , then the difference between the rank of the linguistic label for X2 , cðX2 Þ, is defined as cðX2 Þ ¼ j i:
ð4:4Þ
Similarly, cðX1 Þ, cðX3 Þ, cðX4 Þ, cðX5 Þ, cðX6 Þ and cðX7 Þ are associated with flexibility dimensions X1 , X3 , X4 , X5 , X6 and X7 , respectively. Comparing the differences of rank for each flexibility dimension may yield maximum positive values and then determine which dimensions of MF represent the best direction for improvement. Step I-3: Second stage assessment. The importance of flexibility dimensions are also linguistic labels, presented in Table 1, established by operation managers. Therefore, both the grade of importance and the first stage aggregate rating on each flexibility dimension should be evaluated to determine the degree of MF. Let ððIAð1Þ ; XAð1Þ Þ; . . . ; ðIAð7Þ ; XAð7Þ ÞÞ be a set of two linguistic labels that specify the grade of importance and the first stage aggregate rating, respectively. The linguistic weighted disjunction operator, proposed by Herrera and HerreraViedma [9] and based on a LOWA operator with a maximum entropy weighting vector, W3 , is used. It is obtained from the non-decreasing fuzzy linguistic quantifier Q3 , representing the fuzzy majority over the flexibility dimensions, as follows: IðMF1Þ ¼ UQ3 ðIAð1Þ ; . . . ; IAð7Þ Þ;
ð4:5Þ
Now, the direct derivation is considered. This algorithm aggregates the degrees of flexibility determined by individuals and then aggregates them as a final degree. The aggregate of the rates on the n decision makers can be expressed as follows. Step II-1: First stage assessment. As in step I-2 and the Q2 in Section 4, the first stage aggregate ratings of flexibility dimensions can be obtained as follows: XAðj;mÞ ¼ UQ2 ðXðj;m;1Þ ; Xðj;m;2Þ ; Xðj;m;3Þ ; Xðj;m;4Þ Þ for j ¼ 1; 2; . . . ; n; m ¼ 1; 2; . . . ; 7:
ð5:1Þ
Step II-2: Second stage assessment. As in step I3 and the Q3 in Section 4, the degree of MF for each Dj can be obtained as follows: IAðjÞ ¼ UQ3 ðIAðj;1Þ ; IAðj;2Þ ; . . . ; IAðj;7Þ Þ for j ¼ 1; 2; . . . ; n;
ð5:2Þ
XAðjÞ ¼ MaxðMinðIAðj;1Þ ; XAðj;1Þ Þ; . . . ; MinðIAðj;7Þ ; XAðj;7Þ ÞÞ for j ¼ 1; 2; . . . ; n:
ð5:3Þ
Step II-3: As in step I-3 and the Q1 , in Section 4, to obtain IðMF2Þ ¼ UQ1 ðIAð1Þ ; IAð2Þ ; . . . ; IAð7Þ Þ;
ð5:4Þ
DðMF2Þ ¼ MaxðMinðIAð1Þ ; XAð1Þ Þ; . . . ; MinðIAðnÞ ; XAðnÞ ÞÞ:
ð5:5Þ
The linguistic labels, IðMF2Þ and DðMF2Þ, refer to the importance of MF and the degree of MF, respectively, as assessed by n decision makers. Thus, the difference between IðMF2Þ and DðMF2Þ, cðMF2Þ, is also computed. Whether operation managers must improve MF is thus determined.
DðMF1Þ ¼ MaxðMinðIAð1Þ ; XAð1Þ Þ; . . . ; MinðIAð7Þ ; XAð7Þ ÞÞ; ð4:6Þ
where ÔMinÕ stands for minimization operator. The linguistic labels, IðMF1Þ and DðMF1Þ, represent the importance of MF and the degree of MF, respectively, according to the assessments of n decision makers. Thus, the difference between IðMF1Þ and DðMF1Þ, cðMF1Þ, is also computed. Whether operation managers must improve MF is thus determined.
6. Numerical examples Example 1. Consider the nine-term set (Table 1), and a situation in which four decision makers use linguistic labels, D1 ¼ ½L;
D2 ¼ ½ML;
D3 ¼ ½M;
D4 ¼ ½L:
The fuzzy linguistic approach is applied, using the linguistic fuzzy quantifier ÔmostÕ with the pair
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ð0:3; 0:8Þ; then the group linguistic label can be obtained as follows. The algorithm for calculating the maximum entropy weights of LOWA operator (see the Appendix A) yields the weights W , the orness measure a and the maximum entropy weights W as follows: W ¼ ½0
0:4
0:5
0:1
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a ¼ ðð4 1Þ 0 þ ð4 2Þ 0:4 þ ð4 3Þ 0:5 þ ð4 4Þ 0:1Þ=ð4 1Þ ¼ 0:4333; W ¼ ½ 0:1932
0:2269
0:2666
0:3133
in which the following algebraic equation can be obtained: ð1 0:433Þ h3 þ ð2=3 0:433Þ h2
in which
þ ð1=3 0:433Þ h ¼ 0:
W ð2Þ ¼ Qð2=4Þ Qð1=4Þ
Its positive solution is h ¼ 0:8511, and associated parameter b ¼ ð4 1Þ ln 0:8511 ¼ 0:4835,
¼ ðð0:5 0:3Þ=ð0:8 0:3ÞÞ 0 ¼ 0:4;
Table 4 The fuzzy linguistic quantifiers, grades of importance and ratings of perfonnance of three systems for three decision makers Flexibility dimension (most)
Flexibility metric (most)
Grade of importance D1
D2
D3
Rating of performance D1 I
II
III
I
II
III
I
II
III
X1
X11 X12 X13 X14
H
VH
H
M MH MH M
L VL L VL
H DH H H
M M MH ML
L L VL VL
H VH VH VH
MH ML M MH
L VL DL VL
VH DH H VH
X2
X21 X22 X23 X24
VH
VH
DH
ML ML M ML
DL VL VL L
VH DH DH VH
MH ML M M
DL L DL VL
VH VH DH H
M M ML MH
VL VL L VL
H VH DH H
X3
X31 X32 X33 X34
H
MH
MH
M ML ML MH
VL DL VL VL
H VH VH VH
M MH M MH
VL VL L DL
VH H DH VH
ML MH ML ML
VL DL VL VL
VH H VH DH
X4
X41 X42 X43 X44
MH
DH
MH
MH M MH MH
DL DL VL VL
H DH H VH
M ML M MH
DL VL L VL
H DH H DH
MH MH M M
DL VL L DL
H VH VH DH
X5
X51 X52 X53 X54
DH
VH
MH
ML M ML ML
L L VL L
DH VH H VH
M M ML M
L VL DL L
VH H VH VH
ML MH M ML
VL VL VL L
VH H H VH
X6
X61 X62 X63 X64
H
VH
VH
MH MH M M
L DL DL VL
VH VH H VH
MH MH ML M
VL VL DL L
H VH H VH
M ML M ML
L DL DL L
H VH VH DH
X7
X71 X72 X73 X74
DH
M
MH
M MH ML M
VL VL VL DL
H H H VH
MH MH ML M
DL VL L DL
VH H H VH
ML ML M ML
DL L L VL
VH VH VH H
D2
D3
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e0:4835ðð42Þ=ð41ÞÞ W ð2Þ ¼ P 0:4835ðð4jÞ=ð41ÞÞ ¼ 0:2269; je
The algorithms proposed in Sections 4 and 5 yield degrees in Table 5. Tables 4 and 5 show that higher the combination of rating of performance and grade of importance the higher the degree of flexibility.
j ¼ 1; 2; 3; 4: Then we have U‘most’ ðL;ML;M;LÞ
2
M
3
6 ML 7 6 7 ¼ W BT ¼ ½ 0:1932 0:2269 0:2666 0:3133 6 7 4 L 5 L ¼ C 4 fð0:1932;MÞ;ð0:2269;MLÞ;ð0:2666;LÞ;ð0:3133;LÞg ¼ 0:1932 M ð10:1932Þ C 3 fð0:2813;MLÞ;ð0:3304;LÞ;ð0:3883;LÞg ¼ S2 ¼ L;
C 3 ) 0:2813 ML ð1 0:2813Þ C 2 fð0:4598; LÞ; ð0:5402; LÞg ¼ S2 ¼ L; C 2 ) 0:4598 L ð1 0:4598Þ L ¼ S2 ¼ L; * k ¼ minf8; 2 þ roundð0:4598 ð2 2ÞÞg ¼ 2: Therefore, L is the group linguistic label. Example 2. Suppose a group has three decision makers, D1 , D2 and D3 , each of whom assesses the flexibility of three manufacturing systems, (I), (II) and (III) using the linguistic labels in Table 1. In order to show the monotonic property, paper suppose that each decision maker assigns a fixed linguistic quantifier to the corresponding flexibility dimensions and metrics. Table 4 presents the relevant data.
Table 5 The resulted degrees of example System (I)
System (II)
System (III)
Algorithm I IðMF1Þ DðMF1Þ cðMF1Þ
MH ML þ2
MH DL þ5
MH H 1
Algorithm II IðMF2Þ DðMF2Þ cðMF2Þ
MH ML þ2
MH VL þ4
MH MH 0
7. Conclusions The proposed methods, using a Ôfuzzy linguistic approachÕ have the advantages of directly acting on linguistic labels, computing results as linguistic labels, and not producing ÔNot a NumberÕ. They are appropriate for situations in which information may be qualitative, or the precise quantitative information is unavailable or the cost of its computation is too high. However, the methods are limited in that they use approximate reasoning, evaluators must perfectly distinguish the set of labels under a similar conception, and must use linguistic labels to express their opinions. The above model with the group decisionmaking structure, used to evaluate the degree of MF, is very useful in manufacturing system development. The grades of importance or ratings of performance must be improved until acceptable when evaluating the degree of MF. If the degree of MF is too low, it may have to be improved. The dimensions of MF on which improvements must best be made should be determined. The model described in this study to evaluate the degree of MF involves group decision-making, and therefore the final value are more objective and unbiased than those individually assessed. Issues of practical importance follow. (1) In general, if an operation manager wants to estimate the degree of flexibility in a manufacturing system, he/she must be invited to participate in a group of evaluators whose collective experience extends across a broad range of manufacturing organizations. Their inputs should be reasonable and unambiguous. (2) Measuring MF is strategically important, and must affect the formation of manufacturing strategy, to ensure that a manufacturing system can cope with environmental changes.
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Appendix A. Algorithm for calculating the maximum entropy weights of LOWA operator [4,11, 13,22] Step 1 : Determine the non-decreasing proportional fuzzy linguistic quantifier Q, used to represent the fuzzy majority over dimensions or metrics, as follows: 8 if r < a; b; with a; b; r 2 ½0; 1. Some non-decreasing proportional linguistic fuzzy quantifiers are typified by terms ÔmostÕ, Ôat least halfÕ, Ôas many as possibleÕ, for example. Respective parameters ða; bÞ are ð0:3; 0:8Þ, ð0; 0:5Þ and ð0:5; 1Þ, respectively. Step 2 : Compute the weights W as follows: wi ¼ Qði=nÞ Qðði 1Þ=nÞ;
i ¼ 1; 2; . . . ; n:
Step 3 : Compute the orness measure a as follows: X , a¼ ðn: iÞwi ðn 1Þ; i ¼ 1; 2 . . . ; n: i
Step 4 : Compute the maximum entropy weights W , which are used in LOWA operator, according to the two-step process. 4-1: Find a positive solution h of the algebraic equation X ððn iÞ=ðn 1Þ aÞhðniÞ ¼ 0; i ¼ 1; 2; . . . ; n: i
4-2: Obtain W from the following equation, using b ¼ ðn 1Þ ln h :
wi
eb ððniÞ=ðn1ÞÞ ¼ P b ððnjÞ=ðn1ÞÞ ; je i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; n:
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[19] N. Slack, The flexibility of manufacturing systems, International Journal of Operations and Production Management 7 (4) (1987) 35–45. [20] D.M. Upton, The management of manufacturing flexibility, California Management Review (1994) 72–89. [21] R.J. Vokurka, S.W. OÕLeary-Kelly, A review of empirical research on manufacturing flexibility, Journal of Operations Management 18 (2000) 485–501.
[22] R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Transactions on Systems, Man and Cybernetics 18 (1988) 183–190. [23] L.A. Zadeh, Fuzzy set, Information and Control 8 (1965) 338–353. [24] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Information Sciences 8 (1975) 199–249, 301–357; 9 (1976) 43–58.
Decision Aiding
Group decision-making using a fuzzy linguistic approach for evaluating the flexibility in a manufacturing system Reay-Chen Wang *, Shian-Jong Chuu Department of Industrial Management, National Taiwan University of Science and Technology, 43, Keelung Road, Section 4, Taipei (106), Taiwan, ROC Received 30 July 2001; accepted 18 September 2002
Abstract This study is to build a group decision-making structure model of flexibility in a manufacturing system development. The paper also proposes two algorithms for determining the degree of manufacturing flexibility (MF) in a fuzzy environment using a fuzzy linguistic approach. While evaluating the degree of MF, one may find the need for improving MF, and determine the dimensions of MF as the best direction to improvement until she/he can accept it The results of this study are more objective and unbiased since they are generated by a group of evaluators. Ó 2002 Elsevier B.V. All rights reserved. Keywords: Manufacturing flexibility; LOWA operators; Group decision-making; Linguistic fuzzy quantifier; Maximum entropy weights
1. Introduction Manufacturing environments have changed so fast in recent decades that the strategic function of manufacturing systems have become increasingly important. Flexible manufacturing, computerintegrate manufacturing and Just-in-Time systems depend on flexibility [15]. Flexibility is emerging as a key competitive priority in todayÕs manufacturing systems.
*
Corresponding author. Tel.: +886-2-27376353; fax: +886-227376344. E-mail address: [email protected] (R.-C. Wang).
Generally, manufacturing flexibility (MF) is the ability of a manufacturing system to cope with environmental changes effectively and efficiently. Operation managers must evaluate MF when making capital investment decisions and measuring performance level [6]. MF is a complex, multidimensional and difficult-to-synthesize concept [17], and so the needs of operation managers have not yet been met. Many reviews have considered definitions of MF, requests for MF, classificatory dimensions of MF, measurement of MF, choices for MF, and interpretations of MF [1,3,8,16,17,21]. GerwinÕs [6] conceptual framework concerns the influence of environmental uncertainty on MF. Upton [20] presented a framework for analyzing MF using
0377-2217/$ - see front matter Ó 2002 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(02)00729-4
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different dimensions, each of which at different time intervals and is specified by three elements–– range, mobility and uniformity. Golden and Powell [7] proposed an inclusive definition in which flexibility can be measured by four metrics––efficiency, responsiveness, versatility and robustness. Many researchers have tried objectively to quantify MF. Several efforts are theoretical and involve only two or three dimensions of MF [1]. The importance of dimensions of flexibility and the subjective evaluation of MF have seldom been addressed. Most operation managers cannot provide exact numerical values to express opinions based on human perception: more realistic measurement uses linguistic assessments instead of numerical values [1,6,10,13,21]. In fact metrics can be measured as linguistic labels (terms) such as very high, high, middle, low, and very low. After Zadeh [23] introduced fuzzy set theory to deal with vague problems, linguistic labels have been used in approximate reasoning within the framework of fuzzy set theory [24] to handle the ambiguity in evaluating data and the vagueness of linguistic expression. Normal trapezoidal fuzzy numbers have been used to characterize linguistic terms used in approximate reasoning. Therefore, the purpose of this study was to build a group decision-making structure model of flexibility in a manufacturing system. Two algorithms are proposed to assess the degree of MF in a fuzzy environment using a fuzzy linguistic approach to any phase of the life cycle. Section 2 presents a fuzzy linguistic approach to evaluating flexibility in a manufacturing system development. Section 3 proposes a hierarchical structure model of flexibility in a manufacturing system development. Assume that a group of n decision makers (D1 ; D2 ; . . . ; Dn ) are responsible for assessing the degree of MF for the management. One method aggregates each parameter assessed by an individual, and aggregates the results to determine the final degree of flexibility. Another method aggregates the degrees of flexibility determined by individuals and then aggregates them as a final degree. Sections 4 and 5 separately consider these two algorithms. Finally, Section 6 presents the numerical examples.
2. Fuzzy linguistic approach 2.1. Linguistic assessments The fuzzy linguistic approach assesses linguistic variables using words or sentences of a natural language [24]. This approach is appropriate for some problems in which information may be qualitative, or quantitative information may not be stated precisely, since either it is unavailable or the cost of its determination is excessive, such that an Ôapproximate valueÕ suffices [10]. Most experts cannot give exact numerical values to express their opinions on flexibility metrics. More appropriately, measurements are stated as linguistic assessments rather than numerical values. When applying a fuzzy linguistic approach to measuring MF, only the importance of flexibility dimensions, based on the manufacturing strategy [6] is considered. The flexibility metrics are considered to be equally important, since certain measures of flexibility are still not widely accepted [1,3,7,21]. For example, when determining MF, Ômix flexibilityÕ is frequently an important factor. Therefore, performance should be measured according to each flexibility metric, and the importance of each flexibility dimension should be determined. As mentioned above, the rating of performance and grade of importance should be rated for each item. Therefore, both were scored on a nine-rank scale, as shown in Table 1. Let S ¼ fs0 ; s1 ; . . . ; s8 g be a finite and totally ordered term set on ½0; 1 with an odd cardinal, where the middle label, s4 , represents ÔaverageÕ, and the remaining terms are placed symmetrically around s4 , and exhibit the following properties [12]. 1. The set is ordered: si = sj if i = j. 2. The negation operator is defined as Negðsi Þ ¼ sj such that j ¼ 8 i. 3. The maximization operator is Maxðsi ; sj Þ ¼ si if si = sj . 4. The minimization operator is Minðsi ; sj Þ ¼ si if si 5 sj . The nine linguistic labels in S ¼ fs0 ; s1 ; . . . ; s8 g were specified. The semantics associated with each
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Table 1 Linguistic labels of rating of performance and grade of importance Nine ranks of rating of performance S0 S1 S2 S3 S4 S5 S6 S7 S8
Nine ranks of grade of importance
¼ DL: definitely low ¼ VL: very low ¼ L: low ¼ ML: more or less low ¼ M: middle ¼ MH: more or less high ¼ H : high ¼ VH: very high ¼ DH: definitely high
S0 S1 S2 S3 S4 S5 S6 S7 S8
label si are given as a linear trapezoidal membership function associated with a normal fuzzy number. Not all individuals agree on the same associations between membership functions and linguistic labels. However, this paper consider a situation in which experts can perfectly distinguish among the set of labels under a similar conception, and can use linguistic labels to express their opinions. 2.2. Combining linguistic labels This work applies a linguistic ordered weighted averaging (LOWA) operator, which uses the maximum entropy weights, of linguistic labels by direct computation on labels; that is, the independently of the semantics of the label set. The LOWA operator defined by Herrera et al. [11], is based on the ordered weighted averaging operator defined by Yager [22], which used in the problem of multiple attribute decision making, and on the convex combination of linguistic labels defined by Delgado et al. [2]. The LOWA operator on linguistic labels acts directly on a finite and total ordered term set. The main problem associated with finite term sets is
¼ DL: definitely low ¼ VL: very low ¼ L: low ¼ ML: more or less low ¼ M: middle ¼ MH: more or less high ¼ H : high ¼ VH: very high ¼ DH: definitely high
that the impact of small changes in the weights on the weighed aggregation can be large, especially when the ÔroundÕ operation is used. Furthermore, in some cases, the aggregation obtained by applying the LOWA operator will produce ÔNot a NumberÕ. For example, aggregation of the above four labels, fDL; VL; H ; VHg using the LOWA operator, is attempted. Associating Ôas many as possibleÕ, ÔmostÕ and Ôat least halfÕ, with parameters ða; bÞ of ð0:5; 1Þ, ð0:3; 0:8Þ and ð0; 0:5Þ, respectively, yields the results in Table 2. Therefore, the algorithm to determine the maximum entropy weights is applied to reduce the main problem of applying the LOWA operator. Let fa1 ; a2 ; . . . ; am g be a set of linguistic labels to be aggregated. Then, the LOWA operator UQ with the non-decreasing proportional linguistic fuzzy quantifier, Q, is defined as UQ ða1 ; a2 ; . . . ; am Þ ¼ W BT ¼ C m ðwk ; bk ; k ¼ 1; 2; . . . ; mÞ ¼ w1 b1 ð1 w1 Þ C m1 ðbh ; bh ; h ¼ 2; 3; . . . ; mÞ;
ð2:1Þ
where W ¼ ½w1 ; w2 ; . . . ; wm , is a maximum entropy weighting vector, such that, wi 2 ½0; 1 and
Table 2 Comparison between LOWAðWQ Þ and LOWAðW Þ Fuzzy majority quantifiers (Q)
Weighting vector (WQ )
Orness measure (a)
Maximum entropy weighting vector (W )
LOWA (WQ )
LOWA (W )
As many as possible ð0:5; 1Þ Most ð0:3; 0:8Þ At least half ð0; 0:5Þ
f0; 0; 0:05; 0:5g f0; 0:4; 0:5; 0:1g f0:5; 0:5; 0; 0g
0.1667 0.4333 0.8333
f0:0311; 0:0856; 0:2355; 0:6478g f0:1931; 0:2269; 0:2666; 0:3133g f0:6478; 0:2355; 0:0856; 0:0311g
VL ML NaNa
VL ML H
a
Is ÔNot a NumberÕ.
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P wi ¼ 1 (see the Appendix A), bh ¼ wh = h wh ; h ¼ 2; 3; . . . ; m, B ¼ ½b1 ; b2 ; . . . ; bm is the associated ordered label vector, each element bi 2 B is the ith largest label in the ða1 ; a2 ; . . . ; am Þ, C m is the convex combination operator of m labels, is the general product of a label by a positive real number and is the general addition of labels defined in [2]. If m ¼ 2, then C 2 is defined as
P
i
C 2 fwi ; bi ; i ¼ 1; 2g ¼ w1 sj ð1 w1 Þ si ¼ sk ; ð2:2Þ sj ; si 2 S
ðj = iÞ
such that k ¼ Minð8; i þ roundðw1 ðj iÞÞÞ;
ð2:3Þ
where ÔroundÕ is the usual round operation, and b1 ¼ s j , b2 ¼ s i . If wj ¼ 1 and wi ¼ 0 with i 6¼ j for all i, then the convex combination is defined as C m fwi ; bi ;
i ¼ 1; 2; . . . ; mg ¼ bj :
3. Hierarchical structure model of MF A systematic approach is proposed to evaluate the degree of MF, using a fuzzy linguistic approach and hierarchical structure analysis. This method is suited to decision making in a fuzzy environment. The dimensions of flexibility presented by Gerwin [5,6], Slack [18,19], De Toni and Tonchia [3] and Beach et al. [1] were expressed seven dimensions, including mix flexibility, changeover flexibility, and others, according to the relationship between MF and environmental changes. Furthermore, each dimension was divided into four flexibility metrics [7,14], such as efficiency, responsiveness, versatility and robustness. For convenience, the dimension mix flexibility was represented as X1 , changeover flexibility as X2 , and so on. The metrics oh the mth flexibility dimension were represented as in Fig. 1; for example, efficiency was denoted by Xm1 , responsiveness by Xm2 , and so on. The decision makers consider the grade of importance and related rating of performance,
grading both as S ¼ fs0 ; s1 ; . . . ; s8 g. Suppose a group of n decision makers ðD1 ; D2 ; . . . ; Dn Þ are responsible for assessing the degree of MF. The symbol Iðj;mÞ is used to denote the grade of importance of dimension Xm ; Xðj;m;kÞ the rating of performance of flexibility metric Xmk , according to decision maker Dj Õs assessing data (j ¼ 1; 2; . . . ; n; m ¼ 1; 2; . . . ; 7; k ¼ 1; 2; 3; 4). Table 3 represents the above given the data assessed by decision maker Dj (j ¼ 1; 2; . . . ; n). The data assessed by all n decision makers are combined to evaluate the final degree of MF. As in [10], basically two algorithms are considered. A direct algorithm, fD1 ; D2 ; . . . ; Dn g ) solution is based on the individual preference relations. An indirect algorithm, fD1 ; D2 ; . . . ; Dn g ) D ) solution is based on an aggregated preference relation of the group. Therefore, the following two sections of this paper proposes two algorithms for evaluating the degree of MF for use by a group decision makers. 4. Algorithm I Here, an indirect derivation is considered. This algorithm aggregates each parameter assessed by an individual, and aggregates the results to produce the final degree of flexibility. Step I-1: Let Iðj;mÞ and Xðj;m;kÞ be linguistic labels in S ¼ fs0 ; s1 ; . . . ; s8 g. The aggregated parameters obtained from the n decision makersÕ linguistic data can be expressed as follows: IAðmÞ ¼ UQ1 ðIð1;mÞ ; Ið2;mÞ ; . . . ; Iðn;mÞ Þ m ¼ 1; 2; . . . ; 7;
for
XAðm;kÞ ¼ UQ1 ðmð1;m;kÞ ; mð2;m;kÞ ; . . . ; mðn;m;kÞ Þ m ¼ 1; 2; . . . ; 7; k ¼ 1; 2; 3; 4;
ð4:1Þ for ð4:2Þ
where UQ1 is the LOWA operator with the maximum entropy weights, W1 , obtained from the nondecreasing fuzzy linguistic quantifier Q1 , which represents the fuzzy majority over the n decision makers.
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Fig. 1. Hierarchical structure model of MF.
Table 3 The contents of structure model Flexibility dimension (with Q2 ) Grade of importance Flexibility metric (with Q1 ) Rating of performance
X1
X2
X3
X4
X5
X6
X7
IðX1 Þ
IðX2 Þ
IðX3 Þ
IðX4 Þ
IðX5 Þ
IðX6 Þ
IðX7 Þ
X11 X12 X13 X14
X21 X22 X23 X24
X31 X32 X33 X34
X41 X42 X43 X44
X51 X52 X53 X54
X61 X62 X63 X64
X71 X72 X73 X74
m11 m12 m13 m14
m21 m22 m23 m24
m31 m32 m33 m34
m41 m42 m43 m44
m51 m52 m53 m54
m61 m62 m63 m64
m71 m72 m73 m74
Step I-2: First stage assessment. The rating of performance is rated against flexibility metrics using linguistic labels, as in Table 1, which are determined by operation managers. Using the concept of fuzzy majority over the flexibility met-
rics specified by a fuzzy linguistic quantifier Q2 , and using the LOWA operator associated with the maximum entropy weights, W2 , yields the first stage aggregate ratings on dimensions of flexibility, as follows:
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XAðmÞ ¼ UQ2 ðXAðm;1Þ ; XAðm;2Þ ; XAðm;3Þ ; XAðm;4Þ Þ for m ¼ 1; 2; . . . ; 7:
5. Algorithm II ð4:3Þ
Thus, computing the differences between the ranks of the linguistic labels between the grade of importance and the first stage aggregate rating for each flexibility dimension. If XAð2Þ ¼ si and IAð2Þ ¼ sj , then the difference between the rank of the linguistic label for X2 , cðX2 Þ, is defined as cðX2 Þ ¼ j i:
ð4:4Þ
Similarly, cðX1 Þ, cðX3 Þ, cðX4 Þ, cðX5 Þ, cðX6 Þ and cðX7 Þ are associated with flexibility dimensions X1 , X3 , X4 , X5 , X6 and X7 , respectively. Comparing the differences of rank for each flexibility dimension may yield maximum positive values and then determine which dimensions of MF represent the best direction for improvement. Step I-3: Second stage assessment. The importance of flexibility dimensions are also linguistic labels, presented in Table 1, established by operation managers. Therefore, both the grade of importance and the first stage aggregate rating on each flexibility dimension should be evaluated to determine the degree of MF. Let ððIAð1Þ ; XAð1Þ Þ; . . . ; ðIAð7Þ ; XAð7Þ ÞÞ be a set of two linguistic labels that specify the grade of importance and the first stage aggregate rating, respectively. The linguistic weighted disjunction operator, proposed by Herrera and HerreraViedma [9] and based on a LOWA operator with a maximum entropy weighting vector, W3 , is used. It is obtained from the non-decreasing fuzzy linguistic quantifier Q3 , representing the fuzzy majority over the flexibility dimensions, as follows: IðMF1Þ ¼ UQ3 ðIAð1Þ ; . . . ; IAð7Þ Þ;
ð4:5Þ
Now, the direct derivation is considered. This algorithm aggregates the degrees of flexibility determined by individuals and then aggregates them as a final degree. The aggregate of the rates on the n decision makers can be expressed as follows. Step II-1: First stage assessment. As in step I-2 and the Q2 in Section 4, the first stage aggregate ratings of flexibility dimensions can be obtained as follows: XAðj;mÞ ¼ UQ2 ðXðj;m;1Þ ; Xðj;m;2Þ ; Xðj;m;3Þ ; Xðj;m;4Þ Þ for j ¼ 1; 2; . . . ; n; m ¼ 1; 2; . . . ; 7:
ð5:1Þ
Step II-2: Second stage assessment. As in step I3 and the Q3 in Section 4, the degree of MF for each Dj can be obtained as follows: IAðjÞ ¼ UQ3 ðIAðj;1Þ ; IAðj;2Þ ; . . . ; IAðj;7Þ Þ for j ¼ 1; 2; . . . ; n;
ð5:2Þ
XAðjÞ ¼ MaxðMinðIAðj;1Þ ; XAðj;1Þ Þ; . . . ; MinðIAðj;7Þ ; XAðj;7Þ ÞÞ for j ¼ 1; 2; . . . ; n:
ð5:3Þ
Step II-3: As in step I-3 and the Q1 , in Section 4, to obtain IðMF2Þ ¼ UQ1 ðIAð1Þ ; IAð2Þ ; . . . ; IAð7Þ Þ;
ð5:4Þ
DðMF2Þ ¼ MaxðMinðIAð1Þ ; XAð1Þ Þ; . . . ; MinðIAðnÞ ; XAðnÞ ÞÞ:
ð5:5Þ
The linguistic labels, IðMF2Þ and DðMF2Þ, refer to the importance of MF and the degree of MF, respectively, as assessed by n decision makers. Thus, the difference between IðMF2Þ and DðMF2Þ, cðMF2Þ, is also computed. Whether operation managers must improve MF is thus determined.
DðMF1Þ ¼ MaxðMinðIAð1Þ ; XAð1Þ Þ; . . . ; MinðIAð7Þ ; XAð7Þ ÞÞ; ð4:6Þ
where ÔMinÕ stands for minimization operator. The linguistic labels, IðMF1Þ and DðMF1Þ, represent the importance of MF and the degree of MF, respectively, according to the assessments of n decision makers. Thus, the difference between IðMF1Þ and DðMF1Þ, cðMF1Þ, is also computed. Whether operation managers must improve MF is thus determined.
6. Numerical examples Example 1. Consider the nine-term set (Table 1), and a situation in which four decision makers use linguistic labels, D1 ¼ ½L;
D2 ¼ ½ML;
D3 ¼ ½M;
D4 ¼ ½L:
The fuzzy linguistic approach is applied, using the linguistic fuzzy quantifier ÔmostÕ with the pair
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ð0:3; 0:8Þ; then the group linguistic label can be obtained as follows. The algorithm for calculating the maximum entropy weights of LOWA operator (see the Appendix A) yields the weights W , the orness measure a and the maximum entropy weights W as follows: W ¼ ½0
0:4
0:5
0:1
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a ¼ ðð4 1Þ 0 þ ð4 2Þ 0:4 þ ð4 3Þ 0:5 þ ð4 4Þ 0:1Þ=ð4 1Þ ¼ 0:4333; W ¼ ½ 0:1932
0:2269
0:2666
0:3133
in which the following algebraic equation can be obtained: ð1 0:433Þ h3 þ ð2=3 0:433Þ h2
in which
þ ð1=3 0:433Þ h ¼ 0:
W ð2Þ ¼ Qð2=4Þ Qð1=4Þ
Its positive solution is h ¼ 0:8511, and associated parameter b ¼ ð4 1Þ ln 0:8511 ¼ 0:4835,
¼ ðð0:5 0:3Þ=ð0:8 0:3ÞÞ 0 ¼ 0:4;
Table 4 The fuzzy linguistic quantifiers, grades of importance and ratings of perfonnance of three systems for three decision makers Flexibility dimension (most)
Flexibility metric (most)
Grade of importance D1
D2
D3
Rating of performance D1 I
II
III
I
II
III
I
II
III
X1
X11 X12 X13 X14
H
VH
H
M MH MH M
L VL L VL
H DH H H
M M MH ML
L L VL VL
H VH VH VH
MH ML M MH
L VL DL VL
VH DH H VH
X2
X21 X22 X23 X24
VH
VH
DH
ML ML M ML
DL VL VL L
VH DH DH VH
MH ML M M
DL L DL VL
VH VH DH H
M M ML MH
VL VL L VL
H VH DH H
X3
X31 X32 X33 X34
H
MH
MH
M ML ML MH
VL DL VL VL
H VH VH VH
M MH M MH
VL VL L DL
VH H DH VH
ML MH ML ML
VL DL VL VL
VH H VH DH
X4
X41 X42 X43 X44
MH
DH
MH
MH M MH MH
DL DL VL VL
H DH H VH
M ML M MH
DL VL L VL
H DH H DH
MH MH M M
DL VL L DL
H VH VH DH
X5
X51 X52 X53 X54
DH
VH
MH
ML M ML ML
L L VL L
DH VH H VH
M M ML M
L VL DL L
VH H VH VH
ML MH M ML
VL VL VL L
VH H H VH
X6
X61 X62 X63 X64
H
VH
VH
MH MH M M
L DL DL VL
VH VH H VH
MH MH ML M
VL VL DL L
H VH H VH
M ML M ML
L DL DL L
H VH VH DH
X7
X71 X72 X73 X74
DH
M
MH
M MH ML M
VL VL VL DL
H H H VH
MH MH ML M
DL VL L DL
VH H H VH
ML ML M ML
DL L L VL
VH VH VH H
D2
D3
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e0:4835ðð42Þ=ð41ÞÞ W ð2Þ ¼ P 0:4835ðð4jÞ=ð41ÞÞ ¼ 0:2269; je
The algorithms proposed in Sections 4 and 5 yield degrees in Table 5. Tables 4 and 5 show that higher the combination of rating of performance and grade of importance the higher the degree of flexibility.
j ¼ 1; 2; 3; 4: Then we have U‘most’ ðL;ML;M;LÞ
2
M
3
6 ML 7 6 7 ¼ W BT ¼ ½ 0:1932 0:2269 0:2666 0:3133 6 7 4 L 5 L ¼ C 4 fð0:1932;MÞ;ð0:2269;MLÞ;ð0:2666;LÞ;ð0:3133;LÞg ¼ 0:1932 M ð10:1932Þ C 3 fð0:2813;MLÞ;ð0:3304;LÞ;ð0:3883;LÞg ¼ S2 ¼ L;
C 3 ) 0:2813 ML ð1 0:2813Þ C 2 fð0:4598; LÞ; ð0:5402; LÞg ¼ S2 ¼ L; C 2 ) 0:4598 L ð1 0:4598Þ L ¼ S2 ¼ L; * k ¼ minf8; 2 þ roundð0:4598 ð2 2ÞÞg ¼ 2: Therefore, L is the group linguistic label. Example 2. Suppose a group has three decision makers, D1 , D2 and D3 , each of whom assesses the flexibility of three manufacturing systems, (I), (II) and (III) using the linguistic labels in Table 1. In order to show the monotonic property, paper suppose that each decision maker assigns a fixed linguistic quantifier to the corresponding flexibility dimensions and metrics. Table 4 presents the relevant data.
Table 5 The resulted degrees of example System (I)
System (II)
System (III)
Algorithm I IðMF1Þ DðMF1Þ cðMF1Þ
MH ML þ2
MH DL þ5
MH H 1
Algorithm II IðMF2Þ DðMF2Þ cðMF2Þ
MH ML þ2
MH VL þ4
MH MH 0
7. Conclusions The proposed methods, using a Ôfuzzy linguistic approachÕ have the advantages of directly acting on linguistic labels, computing results as linguistic labels, and not producing ÔNot a NumberÕ. They are appropriate for situations in which information may be qualitative, or the precise quantitative information is unavailable or the cost of its computation is too high. However, the methods are limited in that they use approximate reasoning, evaluators must perfectly distinguish the set of labels under a similar conception, and must use linguistic labels to express their opinions. The above model with the group decisionmaking structure, used to evaluate the degree of MF, is very useful in manufacturing system development. The grades of importance or ratings of performance must be improved until acceptable when evaluating the degree of MF. If the degree of MF is too low, it may have to be improved. The dimensions of MF on which improvements must best be made should be determined. The model described in this study to evaluate the degree of MF involves group decision-making, and therefore the final value are more objective and unbiased than those individually assessed. Issues of practical importance follow. (1) In general, if an operation manager wants to estimate the degree of flexibility in a manufacturing system, he/she must be invited to participate in a group of evaluators whose collective experience extends across a broad range of manufacturing organizations. Their inputs should be reasonable and unambiguous. (2) Measuring MF is strategically important, and must affect the formation of manufacturing strategy, to ensure that a manufacturing system can cope with environmental changes.
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Appendix A. Algorithm for calculating the maximum entropy weights of LOWA operator [4,11, 13,22] Step 1 : Determine the non-decreasing proportional fuzzy linguistic quantifier Q, used to represent the fuzzy majority over dimensions or metrics, as follows: 8 if r < a; b; with a; b; r 2 ½0; 1. Some non-decreasing proportional linguistic fuzzy quantifiers are typified by terms ÔmostÕ, Ôat least halfÕ, Ôas many as possibleÕ, for example. Respective parameters ða; bÞ are ð0:3; 0:8Þ, ð0; 0:5Þ and ð0:5; 1Þ, respectively. Step 2 : Compute the weights W as follows: wi ¼ Qði=nÞ Qðði 1Þ=nÞ;
i ¼ 1; 2; . . . ; n:
Step 3 : Compute the orness measure a as follows: X , a¼ ðn: iÞwi ðn 1Þ; i ¼ 1; 2 . . . ; n: i
Step 4 : Compute the maximum entropy weights W , which are used in LOWA operator, according to the two-step process. 4-1: Find a positive solution h of the algebraic equation X ððn iÞ=ðn 1Þ aÞhðniÞ ¼ 0; i ¼ 1; 2; . . . ; n: i
4-2: Obtain W from the following equation, using b ¼ ðn 1Þ ln h :
wi
eb ððniÞ=ðn1ÞÞ ¼ P b ððnjÞ=ðn1ÞÞ ; je i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; n:
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