O R I G I N A L PA P E R
J.-R. Chang · T.-H. Ho · C.-H. Cheng · A.-P. Chen
Dynamic fuzzy OWA model for group multiple criteria decision
making
Published online: 18 May 2005 © Springer-Verlag 2005
Abstract Obtaining relative weights in MCDM problems is a very important issue. The Ordered Weighted Averaging (OWA) aggregation operators have been extensively adopted to assign the relative weights of numerous criteria. However, previous aggregation operators (including OWA) are inde-pendent of aggregation situations. To solve the problem, this study proposes a new aggregation model – dynamic fuzzy OWA based on situation model, which can modify the asso-ciated dynamic weight based on the aggregation situation and can work like a “magnifying lens” to enlarge the most important attribute dependent on minimal information, or can obtain equal attribute weights based on maximal information. Two examples are adopted in this paper for comparison and showing the effects under different weights.
Keywords Fuzzy multiple criteria decision making (FMCDM)· Ordered weighted averaging (OWA) · Dynamic fuzzy OWA model· Linguistic variable
J.-R. Chang (
B
)· C.-H. ChengDepartment of Information Managements,
National Yunlin University of Science and Technology, 123 University Road, section 3, Touliu, Yunlin 640, Taiwan Tel.: +886-5-5342601
Fax: +886-5-5312077
E-mail: g9120806@yuntech.edu.tw J.-R. Chang
Graduate Institute of Management,
National Yunlin University of Science and Technology, 123 University Road, section 3, Touliu, Yunlin 640, Taiwan T.-H. Ho· A.-P. Chen
Institute of Information Management, National Chiao Tung University,
1001 Ta Hsueh Road, Hsinchu 300, Taiwan T.-H. Ho
Department of Information Management, Nan Jeon Institute of Technology,
178, Zhao-Qin Road, Yenshui, Tainan, Taiwan
1 Introduction
Information aggregation can be applied to many situations, including neural networks, fuzzy logic controllers, expert systems, and multi-criteria decision support systems [14]. In a vague condition, fuzzy set theory [30] can provide an attractive connection to represent uncertain information and can aggregate them properly. The existing aggregation oper-ators are, in general, the t-norm [26], t-conorm [26], mean operators [11], Yager’s operator [28] and γ -operator [32].
Multi-criteria decision making (MCDM) models are char-acterized to evaluate a finite set of alternatives. The main pur-pose of solving MCDM problems is to measure the overall preference values of the alternatives. Two reasons reveal the importance of obtaining relative weights in MCDM prob-lems. First, numbers of approaches have been proposed to assess criteria weights, which are then used explicitly to aggregate specific priority scores [5–7, 17, 19, 23, 24, 27, 31]. Second, some experiments [1, 22, 25] demonstrate that differ-ent approaches for deriving weights may lead to differdiffer-ent results [16].
When an attempt is made to solve the MCDM problem by aggregating the information of each attribute in many disciplines, a problem of aggregating criteria functions to form overall decision functions occurs owing to these cri-teria always being interdependent. One extreme is the sit-uation in which we hope that all the criteria will be sat-isfied (“and” situation), while another situation is the case in which satisfying simple criteria satisfaction is that any of the criteria is all we desire (“or” situation) [27, 29]. In 1988, Yager [27] first introduced the concept of OWA oper-ators to solve this problem. The OWA operoper-ators have the ability to provide an aggregation lying between these two extremes, so it more fit the thought of human being (be-tween the “and” and “or” situations) [27]. O’Hagan [18] is the first to use the concept of entropy in the OWA opera-tion, but situation factor has not yet been taken into the con-sideration of this method. Mesiar and Saminger [15] have shown that in the class of OWA operators is on the domination over the t -norm, and the domination of OWA operators and
related operators over continuous Archimedean t-norms is also discussed.
However, these aggregation operators [8, 11, 28, 32] are independent of their situations and cannot reflect change in situations [4]. To resolve this problem, this study proposes a dynamic OWA aggregation model based on the faster OWA operator, which has been introduced by Fuller and Majlend-er [8] and can work like a magnifying lens and adjust its focus based on the sparest information to change the dynamic attribute weights to revise the weight of each attribute based on aggregation situation, and then to provide suggestions to decision maker (DM).
To verify the proposed model, two examples are adopted in this paper. The first example is to evaluate aggregative risks in software development with three software projects [13]. The results of proposed method will compare with Lee [13] and Chen [2]. The second example is to solve the distri-bution center selection problem in Taiwan [3], and the results will also compare with the Chen’s method [3].
The rest of this paper is organized as follows. Section 2 presents a basic concept of the OWA operator. Section 3 intro-duces the proposed model and a generalized algorithm. The verification and comparison based on experimental results for two examples are introduced in Sect. 4. Subsequently, Sect. 5 discuss the finding and give some suggestions. Conclusions are finally made in Sect. 6.
2 OWA operator
The OWA operator [27, 29] is an important aggregation oper-ator within the class of weighted aggregation methods. Many related studies have been conducted in recent years. For exam-ple, Fuller and Majlender [8] have used Lagrange multipliers to derive a polynomial equation to solve constrained opti-mization problem and to determine the optimal weighting vector. Meanwhile, Smolikova and Wachowiak [23] have described and compared aggregation techniques for expert multi-criteria decision-making method. Furthermore, Ribeiro and Pereira [20] have presented an aggregation schema based on generalized mixture operators using weighting functions, and have compared it with these two standard aggregation method: weighting averaging and ordered weighted averag-ing in the context of multiple attribute decision makaverag-ing. The main concepts of this approach are derived from the OWA operators of Yager [27] and Fuller and Majlender [8]. This section introduces the main content of their methods.
2.1 Yager’s OWA
According to previous studies, t -norm and t-conorm are based on the theory of logic [26], and the mean operators [11] are based on the mathematical properties of averaging. However, in the opinion of Choi [4], these types of aggregation oper-ators are independent of aggregation situation. Even though
Yager’s operator [28] and γ -operator [32] are suggested as an aggregation method using parameter, at present, the defi-nition of such a parameter is still missing [4].
Yager [27] proposed an OWA operator, which had the ability to get optimal weights of the attributes based on the rank of these weighting vectors after aggregation process (reference to Definition 1).
Definition 1. An OWA operator of dimension n is a map-ping F : Rn →R, that has an associated weighting vector W = [w1, w2, . . . , wn]Tof having the properties
i
wi = 1, ∀wi ∈ [0, 1], i = 1, . . .., n, and such that f (a1,... .,an) =
n
j=1
wjbj (1)
where bj is the j th largest element of the collection of the aggregated objects{a1,... .,an}
Yager [27, 29] also introduced two important character-izing measures in respect to the weighting vector W of an OWA operator. The first one was the measure of orness of the aggregation, which was defined as
Orness(W )= 1 n − 1 n i=1 (n − i) wi (2)
And, the second one, implying the measure of dispersion of the aggregation, was defined as
Disp(W )= − n
i=1
wi ln Wi. (3)
And it measures the degree to which W takes into account all information in the aggregation.
O’Hagan [18] suggested a method which combines the principle of maximum entropy [9, 10, 12, 21] and Yager’s ap-proach [27] to determine a special class of OWA operators having the maximal entropy of the OWA weights for a given level of orness. This approach was based on the solution of the following problem:
Maximize the function − n
i=1
wiln Wi Subject to the constraints α= 1
n − 1 n i=1 (n − i) wi, 0≤ α ≤ 1 i wi = 1, ∀wi ∈ [0, 1], i = 1, . . .., n, (4)
2.2 Fuller and Majlender’s OWA
Fuller and Majlender [8] used the method of Lagrange mul-tipliers to transfer Yager’s OWA equation to a polynomial equation, which can determine the optimal weighting vector. By their method, the associated weighting vector is easily obtained by (5)–(7). ln wj = j − 1 n − 1ln wn+ n − j n − 1ln w1 ⇒ wj = n−1 w1n−jwnj−1, (5) and wn= ((n − 1)α − n) w1+1 (n − 1)α + 1 − n w1 , (6) then w1[(n− 1) α + 1 − n w1]n = [(n − 1) α]n−1[((n− 1) α − n) w1+1] . (7) So the optimal value of w1should satisfy equation (7). When
w1is computed, we can determine wnfrom equation (6) and
then the other weights are obtained from equation (5). In a special case, when w1 = w2 = · · · = wn = 1/n ⇒
disp(W )= ln n which is optimal solution to equation (5) for α = 0.5.
3 New dynamic OWA aggregation model 3.1 Dynamic fuzzy OWA model
After comparing the operators in [4, 8, 11, 20, 23, 27, 28, 32] with the OWA operator, the paper finds that the OWA oper-ator has the rational aggregation result, and more closely fits the thoughts of human beings (between the “and” and “or” situations) [27]. Moreover, under the circumstances of maximal information entropy, the OWA operator can get the optimum result of the aggregation. However, it lacks the abil-ity to reflect the aggregative situation during the aggregation process because the previous OWA operators use a common parameter (i.e. α), but do not view it as the situational factor. To maintain the useful character of the OWA (rational aggre-gation result) and correct the shortcomings (lack to reflect the aggregative situation), this study adds two main concepts (see Fig. 1):
(1) Facilitating dynamic aggregation result (attribute wei-ghts) by feedback.
(2) Changing the attribute weights based on situation (with situation parameter α).
After joining these two characters with the fundamental OWA aggregation model, this work proposes a new fuzzy OWA aggregation model, and clarifies the main differences be-tween the proposed model and other aggregation methods in Table 1. This new model not only has the ability to mod-ify forecasting results of functions corresponding with the aggregative situation, but also can obtain associated attribute
weights that rely on the OWA operator matching the model of human thoughts.
The first concept of modifying the aggregation dynamic attribute weights is the process, which is given to experts who want to evaluate the projects different weights. In this way, the experts will have different affects on integral result after evaluation. For example, if the evaluative time is regarded as a criterion to measure the degree of information quality, the newly-coming experts will be assigned a higher weight. This step can enable the newly-coming experts to have more influence on the attribute weights and individual project eval-uation. Consequently, different attribute ratings can obtain dissimilar attribute weights and also different final proposed solutions for reference by decision makers.
Second, the concept of changing weights of each attribute “based on situation” is that the decision maker (or project manager) determines what is the value of parameter α from information entropy of actual aggregative situation. There-fore, the proposed model can be used to obtain attribute weights by rating them after OWA aggregation according to α. The main advantage of this concept is that the model can be treated as a magnifying lens to determine the most important attribute (assign weight = 1) based on the sparest information (i.e. optimistic and α = 0 or 1) situation. On the other hand, when α= 0.5 (moderate situation), the pro-posed model can obtain attribute weights (equal weights of attributes) based on maximal information.
3.2 Algorithm for the proposed model
The steps of proposed algorithm are as follows:
Step 1 Build hierarchical structure model from
determina-tion problem and number (N) of attributes/criteria.
Step 2 Obtain opinions of domain experts and then collect
their evaluative attribute weights of attributes in re-spect to the hierarchical structure model.
Step 3 List the feasible projects/alternatives, and request the
experts to evaluate the grades of these projects.
Step 4 If no new expert is available, execute step 5. If the
ex-perts do not have significant orderings, assign equal weight for evaluation. Otherwise, perform the OWA aggregation process to obtain the weights of experts for evaluation.
Step 5 The weights of each expert multiply their evaluative
attribute weights to form the aggregative weights of attributes.
Step 6 Sort the attribute weights and execute OWA
aggrega-tion (by equaaggrega-tions (5)–(7)) to obtain refined attribute weights.
Step 7 If the aggregative weights of sub-attributes are exist,
to distribute the refined weight(s) of the sub-attri-butes of each attribute based on the ratio of weights of these sub-attributes given by the experts.
Step 8 Multiply the weights of the attributes by their
pro-ject grades, and then rank their orderings to make reference solution to the decision maker.
Fig. 1 A schematic view of proposed model
Table 1 Main differences between proposed model and other aggregation methods
Yager’s Fuller and Majlender’s Choi’s Lee’s two Chen’s Chen’s [3] Proposed OWA [27] OWA [8] operator [4] algorithm [13] algorithm [2] algorithm dynamic OWA
Aggregation operator Yes Yes Yes No No No Yes
Situation parameter (α) Partial∗ Partial∗ Yes No No No Yes
Feedback No No No No No No Yes
Fuzzy Input No No No Yes Yes Yes Yes
∗The OWA operators use a common parameter (i.e. α), but they do not view it as the situational factor.
4 Verification and comparison
In this section, two examples are adopted to illustrate and ver-ify the proposed model: (1) evaluating the risks of software development; (2) external evaluation of distribution centers in logistic. The results and comparisons of each example are also described.
4.1 Example I: evaluation of software development risk To compare the fuzzy group decision making methods, this section first introduces two algorithms developed by Lee for group decision making structure model of risk in software development, and then presents the algorithm of Chen. We also adjust the proposed algorithm in Sect. 4.1.3 to fit this example.
4.1.1 Lee’s [13] algorithms
Lee [13] assumes that there is a group of n experts (D1,D2,
. . .., Dn) to assess the risks for a project in software develop-ment. Let the symbol ˜W2(j,h)denote the relative importance
weight given by expert Dj to attribute h, and let ˜W1(j,h,k), ˜r(j,h,k)and ˜i(j,h,k)denote the weight, the grade of risk, and
the grade of importance given to the risk item Xhkfor expert Dj’s assessment data (j = 1, 2,. . ., n; h = 1, 2, . . . , 6; k = 1, 2, . . ., n(h), and n(h) is the number of risk items for attributes Xh), and (+) and (×) denote the addition and multiplication operators of triangular fuzzy numbers (TFNs), respectively. Table 2 shows an example of the contents of the hierarchical structure model, where
˜ W2(j,h)= (a2(j,h), b2(j,h), c2(j,h)) (8) ˜ W1(j,h,k)= (a1(j,h,k), b1(j,h,k), c1(j,h,k)) (9) ˜r(j,h,k)= (a3(j,h,k), b3(j,h,k), c3(j,h,k)) (10) ˜i(j,h,k)= (a4(j,h,k), b4(j,h,k), c4(j,h,k)) . (11)
1. Lee’s first algorithm (Algorithm I): This algorithm aver-ages each parameter individually and then aggregates the results to produce a final rate of aggregative risk. The main context of the algorithm is as follows:
i. Calculate the average of each parameter on the fuzzy data of n decision makers.
ii. First-stage assessment. Establish a fuzzy assess-ment matrix ˜M(Xh) for each attribute Xh, and then use these fuzzy assessment matrices to evaluate the first-stage aggregative assessment matrix ˜R1(h)for
Table 2 The contents of the hierarchical structure model for decision maker Dj[13]
Attribute Risk item Weight-2 Weight-1 Grade of risk Grade of importance
X1 W2(j,1)˜ X11 W1(j,1,1)˜ ˜r(j,1,1) ˜i(j,1,1) X2 W2(j,2)˜ X21 W1(j,2,1)˜ ˜r(j,2,1) ˜i(j,2,1) X22 W1(j,2,2)˜ ˜r(j,2,2) ˜i(j,2,2) X23 W1(j,2,3)˜ ˜r(j,2,3) ˜i(j,2,3) X24 W1(j,2,4)˜ ˜r(j,2,4) ˜i(j,2,4) X3 W˜2(j,3) X31 W1(j,3,1)˜ ˜r(j,3,1) ˜i(j,3,1) X32 W1(j,3,2)˜ ˜r(j,3,2) ˜i(j,3,2) X4 W2(j,4)˜ X41 W1(j,4,1)˜ ˜r(j,4,1) ˜i(j,4,1) X42 W1(j,4,2)˜ ˜r(j,4,2) ˜i(j,4,2) X43 W1(j,4,3)˜ ˜r(j,4,3) ˜i(j,4,3) X44 W˜1(j,4,4) ˜r(j,4,4) ˜i(j,4,4) X5 W2(j,5)˜ X51 W1(j,5,1)˜ ˜r(j,5,1) ˜i(j,5,1) X52 W1(j,5,2)˜ ˜r(j,5,2) ˜i(j,5,2) X6 W2(j,6)˜ X61 W1(j,6,1)˜ ˜r(j,6,1) ˜i(j,6,1)
iii. The second-stage assessment. Calculate the aver-age defuzzified value of W2A(h) (denoted as
GW2A(h)).
iv. Calculate the final rate of aggregative risk RIK by the centroid defuzzified method as follows:
RIK1= n(DM) j=1 GV (j) × R2(j ) = j h k GV (j) × GW2A(h) 6 q=1GW2A(q) GW1A(h,k) n(h) q=1GW1A(h,q) ×V (h, k, j) (12)
2. Lee’s second algorithm (Algorithm II): This algorithm averages the rate individually and then averages the re-sults to produce a final rate of aggregative risk. Algorithm II is very similar to the above algorithm. This algorithm is further detailed below:
i. Calculate the rate of each project on the n decision makers’ fuzzy data on each parameter.
ii. The first-stage assessment. Establish a fuzzy assess-ment matrix for each attribute Xh, and then use these fuzzy assessment matrices to evaluate the first-stage aggregative assessment matrix ˜R1(h)for
each attribute Xh.
iii. The second-stage assessment. Calculate the aver-age defuzzified value ofW2A(h)(denoted as GW2A(h)).
iv. Evaluate the rate of aggregative risk for decision makers (Dj) first by RIK(j )= 7 m=1GV (m)R2(j,m) 7 q=1R2(j,q) = 7 m=1 GV (m)R2(j,m) (13) Then average then, to obtain
RIK2= 1 n n j=1 RI K(j) = n1 m j h k GV (m) × GW2(j,h) 6 q=1GW2(j,q) GW1(j,h,k) n(h) q=1GW1(j,h,q) ×V (j, h, k, m) (14) 4.1.2 Chen’s [2] algorithms
Chen [2] proposed a new algorithm to evaluate the rate of aggregative risk in software development under a fuzzy group decision making environment. Chen also stated that this algo-rithm has the following advantages: (1) It does not need to form the fuzzy assessment matrices for attributes. (2) It does not need to perform complicated defuzzification operations of fuzzy numbers using the centroid method. This algorithm involves the following steps:
i. Chen first used a defuzzification method of trapezoidal fuzzy numbers to get the defuzzified value (denoted as e) of trapezoidal fuzzy numbers ( ˜M).
ii. Chen used the defuzzified method to convert fuzzy num-ber representations of the weights, the grades of risk, and the grades of importance of risk items into real values.
Table 3 Linguistic values of grades of risk Eleven ranks of grade of risk
Notation Linguistic Value TFNs (˜r(j,h,k))
DL Definitely low (0.0, 0.0, 0.1) EL Extra low (0.0, 0.1, 0.2) VL Very low (0.1, 0.2, 0.3) L Low (0.2, 0.3, 0.4) SL Slightly low (0.3, 0.4, 0.5) M Middle (0.4, 0.5, 0.6) SH Slightly high (0.5, 0.6, 0.7) H High (0.6, 0.7, 0.8) VH Very high (0.7, 0.8, 0.9) EH Extra high (0.8, 0.9, 1.0) DH Definitely high (0.9, 1.0, 1.0)
Table 4 Linguistic values of relative importance Five grades of relative importance
Notation Linguistic Value TFNs ( ˜W2(j,h)/ ˜W1(j,h,k))
1. VL Very low (0.0, 0.0, 0.25)
2. L Low (0.0, 0.25, 0.5)
3. M Middle (0.25, 0.5, 0.75)
4. H High (0.5, 0.75, 1.0)
5. VH Very high (0.75, 1.0, 1.0)
iii. Chen calculated average defizzified values of these real values for decision makers, and also calculated absolute weights of the risk items for each attribute.
iv. Finally, Chen calculated the final rate of aggregative risk RIK of the project by aggregating the risks of each attri-bute.
4.1.3 Adjusted algorithm for Example I
In [13], Lee uses 11 linguistic values for ranking the grades of risk items (see Table 3), which are represented by triangu-lar fuzzy numbers. Furthermore, Lee also allows the experts Table 5 The weights of three projects for two decision makers [13]
Attribute Risk item Weight-2 Weight-1
D1 D2 D1 D2 X1 (0.1, 0.25, 0.35) (0.15, 0.3, 0.5) X11 (0.7, 0.85, 1) (0.8, 0.9, 1) X2 (0.3, 0.5, 0.6) (0.25, 0.4, 0.6) X21 (0.2, 0.3, 0.4) (0.2, 0.35, 0.45) X22 (0.3, 0.4, 0.5) (0.1, 0.3, 0.5) X23 (0.15, 0.3, 0.4) (0.2, 0.3, 0.4) X24 (0.2, 0.35, 0.5) (0.1, 0.3, 0.5) X3 (0.2, 0.3, 0.4) (0.1, 0.2, 0.3) X31 (0.1, 0.15, 0.25) (0.35, 0.55, 0.85) X32 (0.35, 0.5, 0.6) (0.25, 0.55, 0.65) X4 (0.2, 0.35, 0.5) (0.2, 0.3, 0.4) X41 (0.3, 0.4, 0.5) (0.2, 0.3, 0.4) X42 (0.1, 0.2, 0.3) (0.1, 0.2, 0.3) X43 (0.2, 0.3, 0.4) (0.2, 0.3, 0.4) X44 (0.2, 0.3, 0.4) (0.2, 0.3, 0.4) X5 (0.1, 0.25, 0.4) (0.1, 0.3, 0.4) X51 (0.5, 0.6, 0.7) (0.4, 0.5, 0.6) X52 (0.4, 0.5, 0.6) (0.6, 0.7, 0.8) X6 (0.1, 0.3, 0.5) (0.15, 0.3, 0.5) X61 (0.8, 0.9, 1) (0.9, 1, 1)
to use five linguistic values {i.e. VL, L, M, H, VH} (see Table 4) represented by triangular fuzzy numbers (TFNs) for accessing the grades of importance of the risk items.
Decision makers can use either the importance set W ={VL, L, M, H, VH}with the grade set S ={VL, L, SL, M, SH, H, VH}or directly rating by normal triangular fuzzy numbers to access attribute weights, weights of risk items, and grades of risks.
To verify and compare the model with other methods, the result obtained here is compared with that obtained by Lee’s algorithm [13] and Chen’s algorithm [2] to validate the accuracy of the proposed model. Table 1 introduces the differences between their methods and the proposed model, and Sects. 4.1.1 and 4.1.2 describe these three algorithms.
The weights of MCDM, which are obtained by the pro-posed model, will be revised when a new expert joins or a decision maker chooses a different α value to fit the current situation. To verify the proposed model, this study assumes that a symbol D( ˜A) denotes the defuzzification result of this fuzzy number ˜A by the centroid method [13], and uses the symbol (Table 2) in the research of Lee [13] as an example to explain each step. The adjusted steps based on section 3.2’s algorithm are as follows:
Step 1 Build hierarchical structure model from
determina-tion problem and number of attributes (N).
For example, Lee [13] presented the hierarchical structure model of aggregative risk along with attri-butes N = 6.
Step 2 Obtain opinions of experts in software development,
and then collect their evaluative attribute weights in respect to the hierarchical structure model (see Table 5).
Step 3 List the feasible projects, and request the experts to
evaluate the grades of these projects in respect to the risk items (see Table 6).
Table 6 The grades of risk of three projects for two decision makers [13]
Attribute Risk item Grade of risk
D1 D2 X1 X11 (I) (0.4, 0.5, 0.6) (0.5, 0.6, 0.7) (II) (0.6, 0.7, 0.8) (0.7, 0.8, 0.9) (III) (0, 0.1, 0.2) (0, 0, 0.1) X2 X21 (I) (0.2, 0.3, 0.4) (0.2, 0.3, 0.4) (II) (0.6, 0.7, 0.8) (0.8, 0.9, 1) (III) (0.1, 0.2, 0.3) (0.2, 0.3, 0.4) X22 (I) (0.3, 0.4, 0.5) (0.2, 0.4, 0.6) (II) (0.5, 0.6, 0.7) (0.5, 0.6, 0.7) (III) (0, 0, 0.1) (0, 0.1, 0.2) X23 (I) (0.2, 0.4, 0.5) (0.3, 0.5, 0.6) (II) (0.7, 0.8, 0.9) (0.7, 0.8, 0.9) (III) (0.1, 0.2, 0.3) (0, 0.1, 0.2) X24 (I) (0.5, 0.6, 0.7) (0.1, 0.2, 0.3) (II) (0.6, 0.7, 0.8) (0.8, 0.9, 1) (III) (0.1, 0.2, 0.3) (0.2, 0.3, 0.4) X3 X31 (I) (0.25, 0.35, 0.45) (0.35, 0.45, 0.55) (II) (0.7, 0.8, 0.9) (0.7, 0.8, 0.9) (III) (0.1, 0.2, 0.3) (0.1, 0.2, 0.3) X32 (I) (0.4, 0.6, 0.8) (0.2, 0.4, 0.6) (II) (0.5, 0.6, 0.7) (0.5, 0.6, 0.7) (III) (0.2, 0.3, 0.4) (0.1, 0.2, 0.3) X4 X41 (I) (0.2, 0.3, 0.4) (0.25, 0.4, 0.55) (II) (0.6, 0.7, 0.8) (0.7, 0.8, 0.9) (III) (0.2, 0.3, 0.4) (0.2, 0.3, 0.4) X42 (I) (0.1, 0.2, 0.3) (0.2, 0.3, 0.4) (II) (0.6, 0.7, 0.8) (0.7, 0.8, 0.9) (III) (0.1, 0.2, 0.3) (0, 0.1, 0.2) X43 (I) (0.3, 0.4, 0.5) (0.3, 0.4, 0.5) (II) (0.7, 0.8, 0.9) (0.7, 0.8, 0.9) (III) (0.1, 0.2, 0.3) (0.2, 0.3, 0.4) X44 (I) (0.2, 0.3, 0.4) (0.2, 0.3, 0.4) (II) (0.6, 0.7, 0.8) (0.6, 0.7, 0.8) (III) (0.2, 0.3, 0.4) (0.1, 0.2, 0.3) X5 X51 (I) (0.2, 0.3, 0.4) (0.2, 0.3, 0.4) (II) (0.5, 0.6, 0.7) (0.6, 0.7, 0.8) (III) (0.2, 0.3, 0.4) (0.1, 0.2, 0.3) X52 (I) (0.3, 0.4, 0.5) (0.3, 0.4, 0.5) (II) (0.7, 0.8, 0.9) (0.7, 0.8, 0.9) (III) (0.2, 0.3, 0.4) (0.1, 0.2, 0.3) X6 X61 (I) (0.4, 0.5, 0.6) (0.4, 0.5, 0.6) (II) (0.5, 0.6, 0.7) (0.5, 0.6, 0.7) (III) (0.1, 0.2, 0.3) (0.2, 0.3, 0.4)
Step 4 If no new expert is available, execute step 5.
Other-wise, if the experts do not have significant orderings, assign the same weight for evaluation. Meanwhile, perform the OWA aggregation process to obtain the weights for evaluation, and then execute steps 2 and 3 by the result of this step. (This study assumes that
these two experts have equal weights, denoted as
We1=We2=0.5).
Step 5 The weights of each expert are used to multiply their
evaluative attribute weights. The Centroid method [13] then is used to defuzzify the aggregation result of the attribute weights.
This step can be divided into two branch steps——
Step 5.1 Aggregate the attribute weights according to
each expert (The aggregation weight of attri-bute h (denoted as ˜W2A(h)) is calculated by
using equation (15)). Let ˜W2A(h)= 1 n(×) ˜ W2(1,h) (+) ˜W2(2,h)(+)· · ·· · ·, (+) ˜W2(n,h) = (a2A(h), b2A(h), c2A(h)) (15)
where a2A(h)= 1 n n j=1 a2(j,h), b2A(h) = 1n n j=1 b2(j,h), and c2A(h) = 1n n j=1 c2(j,h)
Step 5.2 The Centroid method [13] is used to
defuzz-ify these weights.
The Centroid method is used to defuzzify any kind of fuzzy sets, such as triangular fuzzy numbers, trap-ezoidal fuzzy numbers, and so on. In the present example, only triangular fuzzy number is used. The defuzzified value of the weight of attribute h thus is
DW˜2A(h)
= 1
3× (a2A(h)+ b2A(h)+ c2A(h)) (16)
Step 6 Sort the defuzzified attribute weights and execute
OWA aggregation to obtain refined attribute weights (denoted as W2A(1) ∼ W2A(m) ). The results are shown in Table 7.
This step can be divided into two branch steps—— Step 6.1 Choose an appropriate sorting method to sort
the defuzzified attribute weights.
Step 6.2 Use equations (5)–(7) to obtain the OWA
weights, and replace the attribute weights with these refined weights according to the sorting result. The distribution of the refined weights in Example I under different α’s values is shown in Fig. 2.
Step 7 Like the process of step 5, the aggregative weights
of each risk items are obtained by equation (17) (de-noted as ˜W1A(h,k)), and these aggregative weights are
defuzzified in equation (16) (denoted as D( ˜W1A(1,1)) ∼ D( ˜W1A(6,1))). Equation (19) then is used to
dis-tribute the refined weight(s) of the risk item(s) of each attribute based on the ratio of weights of these risk item(s) given by the experts (denoted as W1A(1,1)
Table 7 The weights of attributes after OWA aggregation
α = 0.5 α = 0.6 α = 0.7 α = 0.8 α = 0.9 α = 1.0
Personnel (W1) 0.16666 0.14614 0.11416 0.07229 0.02548 0
System requirement (W2) 0.16666 0.24676 0.34747 0.47811 0.66372 1
Schedules and budgets (W3) 0.16666 0.10308 0.05437 0.02053 0.00290 0
Developing technology (W4) 0.16666 0.20721 0.23977 0.25473 0.22396 0
External resource (W5) 0.16666 0.12274 0.07876 0.03851 0.00860 0
Performance (W6) 0.16666 0.17401 0.16543 0.13571 0.07559 0
Fig. 2 The distribution of the refined weights in Example I
∼ W
1A(6,1)). The results are shown in Table 8.
Let ˜W1A(h,k) = 1 n(×) ˜ W1(1,h,k)(+) ˜W1(2,h,k) (+)· · ·· · ·, (+) ˜W1(n,h,k)
= (a1A(j,h,k), b1A(j,h,k), c1A(j,h,k)) (17)
wij = wij
jwij × wi, i = number of attributes; j = number of risk items
(18) So, W1A(h,k) = n(h)D( ˜W1A(h,k))
j=1D( ˜W1A(h,j )) ×W 2A(h) (19) where n(h) j=1 D(W1A(h,j )) = 0
Step 8 Multiply the weights of the risk items by their
pro-ject grades, and then rank their orderings to make reference solution to the decision maker.
This step can be divided into two branch steps—— Step 8.1 Calculate the aggregative risk value of each risk
item towards a select project using equation (20).
˜rA(h,k)= 1
n(×)[˜r(1,h,k)
(+)˜r(2,h,k)(+)· · ·· · ·, (+)˜r(n,h,k)]
= (a3A(h,k), b3A(h,k), c3A(h,k)) , (20)
where a3A(h,k)= 1n n j=1 a3(j,h,k), b3A(h,k)=1 n n j=1b3(j,h,k), c3A(h,k)= 1 n n j=1 c3(j,h,k)
Their defuzzified values then are: D˜rA(h,k)= 1
3
a3A(h,k)+ b3A(h,k)+ c3A(h,k)
(21)
Step 8.2 Multiply the refined weights of risk items by
their project grades. Finally, the aggregative re-sult of project P is Agg Result (P ) = n h=1 n(h) k=1 W 1A(h,k) ×D(rA(h,k)) (22)
where P = 1, 2, . . .., n(P ), n is the number of attributes, n(h) is the number of risk items, and n(P ) is the number of projects.
4.1.4 Results for Example I
Because the proposed system has the same results of α = 0.5+ δ and α = 0.5 − δ (0 ≤ δ ≤ 0.5), it merely shows the data of α≥ 0.5 to represent the total result. For example, the aggregation results are the same when α= 0.7 and α = 0.3. Besides, even if parameter α is a continuous value in inter-val [0,1], this study merely illustrates the output inter-values when α = 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0. According to the entropy of information after OWA aggregation of the input data, the output weights of the attributes and risk items are summa-rized in Tables 7 and 8, respectively. From the last column in Table 7, the proposed model can be viewed as a magnifying lens to determine the most important attribute based on the situation of sparest information (i.e. optimistic and α= 0 or 1). In the second column of Table 7, when α = 0.5 (mod-erate situation), the proposed model can obtain the attribute weights (equal weights) based on maximum information.
Similarly, we take α value from 0.5 to 1.0 as the parame-ter for execution according to our algorithm for the purpose of verification in respect to proposed model, and the results are presented below in Table 9.
To verify the validity of the proposed model, this study compares the result of the proposed algorithm with the algo-rithms in Lee [13] and Chen [2]. However, the aggregation results will be changed corresponding with α’s value, thus the extreme values of α are chosen for representation purposes. Therefore, Table 10 selects two extreme points to summa-rize the aggregation results of three projects (i.e. maximum (α=0.5) and minimum information (α=1 or 0) entropy).
Table 10 reveals that the rank of the proposed algorithm is the risk of Project (II) > Project (I) > Project (III), which
Table 8 The weights of risk items after OWA aggregation
Attribute Risk item α = 0.5 α = 0.6 α = 0.7 α = 0.8 α = 0.9 α = 1.0
Personnel
Personnel shortfalls, key person(s) quit (W11) 0.16666 0.14614 0.11416 0.07229 0.02548 0 System requirement
Requirement ambiguity (W21) 0.04112 0.06098 0.08574 0.11797 0.16377 0.24675 Developing the wrong software function (W22) 0.04545 0.06729 0.09476 0.13039 0.18101 0.27272 Developing the wrong user interface (W23) 0.03787 0.05608 0.07897 0.10866 0.15084 0.22727 Continuing stream requirement changes (W24) 0.04220 0.06249 0.08799 0.12108 0.16808 0.25324 Schedules and budgets
Schedule not accurate (W31) 0.07281 0.04503 0.02375 0.00897 0.00126 0 Budget not sufficient (W32) 0.09385 0.05804 0.03061 0.01156 0.00163 0 Developing technology
Gold-plating (W41) 0.05072 0.06306 0.07297 0.07752 0.06816 0 Skill levels inadequate (W42) 0.02898 0.03603 0.04170 0.04430 0.03895 0 Straining hardware (W43) 0.04347 0.05405 0.06255 0.06645 0.05842 0 Straining software (W44) 0.04347 0.05405 0.06255 0.06645 0.05842 0 External resource
Shortfalls in externally furished components (W51) 0.07971 0.05870 0.03767 0.01841 0.00411 0 Shortfalls in externally performed tasks (W52) 0.08695 0.06403 0.04109 0.02009 0.00449 0 Performance
Real-time performance shortfalls (W61) 0.16666 0.17401 0.16543 0.13571 0.07559 0 Table 9 The aggregation result of our algorithm
α = 0.5 α = 0.6 α = 0.7 α = 0.8 α = 0.9 α = 1.0
Project (I) 0.42598 0.41974 0.41098 0.39971 0.38725 0.38668
Project (II) 0.71018 0.71362 0.71848 0.72527 0.73548 0.74545
Project (III) 0.20332 0.20254 0.20208 0.20094 0.19657 0.17727
Table 10 The evaluation results for Example I
Algorithm Project
(I) (II) (III)
Algorithm 1 by Lee [13] 0.19650 0.50917 0.09112 Algorithm 2 by Lee [13] 0.19648 0.50930 0.09173 Algorithm by Chen [2] 0.18541 0.51086 0.05767 Our proposed algorithm (α= 0.5) 0.42598 0.71018 0.20332 Our proposed algorithm (α= 1 or 0) 0.38668 0.74545 0.17727 is the same as the ratings of the projects calculated by the algorithm of Lee [13] and Chen [2]. The proposed model thus can be validated.
4.2 Example II: evaluation of distribution centers in logistic In this section, we adopted an example introduced by Chen [3]. This example discusses the distribution center selection through the evaluation of external performance in one con-venience store of Taiwan.
4.2.1 Chen’s [3] algorithm
The membership function for linguistics values is the same as definition in Table 4. Chen [3] proposed an algorithm of multi-person multi-criteria external performance evaluation in logistics with fuzzy approach can be expressed by the fol-lowing steps:
Step 1 Construction of hierarchical structure
Step 2 Evaluate the importance weight of extracted criterion
(use fuzzy Delphi method)
Step 3 Construction of linguistic scales for linguistic
vari-ables
Step 4 Aggregation of fuzzy appropriateness indices Step 5 Computation of fuzzy overall evaluation Step 6 Defuzzificaion of fuzzy overall evaluation Step 7 Analysis and decision
4.2.2 Adjusted algorithm for Example II
To verify the proposed model, we use the data in Chen [3] as an example to explain each step, and the adjusted steps based on section 3.2’s algorithm are as follows:
Step 1 Build hierarchical structure model from
determina-tion problem and number of attributes (N).
After factor analysis, six criteria were extracted: (1) Efficiency (C1); (2) Customer (C2); (3) Stockouts
(C3); (4) Delivery (C4); (5) Order (C5); (6)
Person-nel (C6). [3]
Step 2 Obtain opinions of domain experts and then collect
their evaluative attribute weights of attributes in re-spect to the hierarchical structure model (see Table 11 [3]).
Step 3 List the feasible alternatives, and request the experts
to evaluate the grades of these projects.
The six commonly used distribution centers in this case are: A1 = Wong Chuan Logistics Corp., A2 =
Da Je Tong Lo-Support International, A3 = Shen
Hong
Logistics Corp., A4 = Retail Support International,
Table 11 The aggregative importance of criteria
Criteria Weight (Triangular Fuzzy Number) Efficiency (C1) W1˜ = (0.2887, 0.2992, 0.3118) Customer (C2) W2˜ = (0.1047, 0.1204, 0.1348) Stockouts (C3) W3˜ = (0.1947, 0.2040, 0.2099) Delivery (C4) W˜4 = (0.0425, 0.0590, 0.1086) Order (C5) W5˜ = (0.0425, 0.0480, 0.0507) Personnel (C6) W6˜ = (0.2325, 0.2639, 0.2975)
Table 12 The fuzzy appropriateness indices of the six alternatives under each criterion [3] Ri Alternatives A1 A2 C1 (0.4168, 0.8025, 0.9714) (0.4011, 0.7814, 0.9571) C2 (0.4156, 0.8000, 0.9506) (0.3950, 0.7731, 0.9256) C3 (0.3756, 0.7444, 0.9500) (0.3688, 0.7344, 0.9506) C4 (0.3600, 0.7225, 0.9750) (0.3913, 0.7663, 0.9750) C5 (0.4200, 0.8075, 1.0000) (0.4550, 0.8550, 1.0000) C6 (0.2875, 0.6188, 0.9250) (0.3538, 0.7163, 0.9500) A3 A4 C1 (0.3893, 0.7643, 0.9650) (0.4211, 0.8086, 0.9786) C2 (0.4044, 0.7856, 0.9500) (0.3994, 0.7775, 0.9381) C3 (0.3606, 0.7231, 0.9256) (0.3913, 0.7663, 0.9506) C4 (0.3913, 0.7663, 0.9750) (0.3763, 0.7450, 0.9500) C5 (0.4200, 0.8075, 1.0000) (0.4200, 0.8075, 1.0000) C6 (0.3413, 0.6975, 0.9500) (0.3263, 0.6763, 0.9500) A5 A6 C1 (0.3939, 0.7707, 0.9650) (0.3896, 0.7646, 0.9507) C2 (0.3800, 0.7519, 0.9381) (0.3931, 0.7713, 0.9375) C3 (0.3675, 0.7331, 0.9506) (0.3375, 0.6906, 0.9506) C4 (0.4075, 0.7888, 0.9750) (0.2975, 0.6350, 0.9250) C5 (0.3850, 0.7600, 0.9500) (0.4200, 0.8075, 1.0000) C6 (0.3263, 0.6763, 0.9250) (0.3138, 0.6575, 0.9250)
Logistics Corp. The grades of these alternatives un-der each criterion are shown in Table 12 [3]. Step 4 If no new expert is available, execute step 5. If the
ex-perts do not have significant orderings, assign equal weight for evaluation. Otherwise, perform the OWA aggregation process to obtain the weights of experts for evaluation.
Step 5 The weights of each expert multiply their
evalua-tive attribute weights to form the aggregaevalua-tive weights of attributes. (In this example, steps 4 and 5 can be
skipped.)
Step 6 Sort the attribute/criteria weights and execute OWA
aggregation (by equation (5)–(7)) to obtain refined attribute weights.
The ranking order of the defuzzified values in Table 11 is ˜W1 > ˜W6 > ˜W3 > ˜W2 > ˜W4 > ˜W5. Then, the
distribution of refined weights for each criterion after OWA is shown in Fig. 3.
Step 7 If the aggregative weights of sub-attributes are exist,
to distribute the refined weight(s) of the sub-attri-butes of each attribute based on the ratio of weights of these sub-attributes given by the experts. (In this
example, this step can be skipped.)
Step 8 Multiply the weights of the attributes by their
pro-ject grades, and then rank their orderings to make
Fig. 3 The distribution of the refined weights in Example II
Fig. 4 The overall results under α= [0.5, 1.0]
reference solution to the decision maker.
The ranking orders and defuzzified values of the fuzzy overall evaluation for each alternative based on the algorithms of proposed model (α= 0.5, 0.6, 0.7, 0.8, 0.9, 1.0) and Chen’s method are shown in Table 13. The overall evaluation values of alternatives under different α’s values is as Fig. 4. The ranking orders of proposed method will be consistent with Chen’s method under α= [0.63, 0.78].
5 Discussion
After the experiments in Sect. 4, we find the proposed model having the following characteristics:
A. Sensitivity and Robustness
Due to the evaluating grades of risk for each project are clearly having trend (i.e. Project (II) > Project (I) > Pro-ject (III) in almost risk items), we find that the ranking order of projects in Example I is robust under different α’s val-ues (see Table 9). However, the ranking order in Example II will change based on different α’s values (see Table 13 and Fig. 4). This is because the evaluating values of criteria for several alternatives are approximate in the in Example II (see Table 12), and the overall evaluating results will easily affect by the refined weights (OWA weights).
B. Effect of “magnifying lens”
From Table 7 and Figs. 2 and 3, if the α value changes from 0.5 to 1.0 (or 0), the weights of attributes will be changed from
Table 13 The evaluation results for Example II
Overall evaluation values Ranking order
A1 A2 A3 A4 A5 A6
Proposed method
α = 0.5 0.7099 0.7240 0.7150 0.7191 0.7052 0.6872 A2>A4>A3>A1>A5>A6 α = 0.6 0.7059 0.7185 0.7099 0.7175 0.7024 0.6851 A2>A4>A3>A1>A5>A6 α = 0.7 0.7039 0.7150 0.7063 0.7177 0.7008 0.6851 A4>A2>A3>A1>A5>A6 α = 0.8 0.7052 0.7140 0.7047 0.7204 0.7011 0.6877 A4>A2>A1>A3>A5>A6 α = 0.9 0.7141 0.7174 0.7075 0.7288 0.7061 0.6954 A4>A2>A1>A3>A5>A6 α = 1.0 0.7483 0.7303 0.7207 0.7542 0.7251 0.7174 A4>A1>A2>A5>A3>A6
Chen’s method [3] 0.6109 0.6236 0.6147 0.6256 0.6089 0.5940 A4>A2>A3>A1>A5>A6
distributed (equal weights) to centralized (the most important attribute).
When dealing with the problems in management, we usu-ally just need to face the key problem, which can help us with overcoming the difficult situation. So, we can use this model to search out the critical attribute of the problem when the α value given by the project manager is 0 or 1 (i.e. under the minimal entropy). Therefore, the proposed model would be a useful tool if the project manager wants to find the most important attribute (or criterion).
After analysis the results of Examples I and II, we suggest the decision maker can adjust α’s value under the following situation:
1. No preference: When a decision maker has no preference toward the criteria, we can assign these attributes equal weight. Under this circumstance, the suggesting α’s value is 0.5.
2. Partial preference: We suggest the range of α’s values is [0.6, 0.9], when a decision maker has collected cri-teria weights from domain experts and want to execute sensitivity analysis for making final decision based the opinions of experts.
3. Single preference: If the decision maker is confident and believe in the most important criterion, we suggest to assign α= 1.0. This can enlarge the effect of this single preference criterion.
6 Conclusion
This study has proposed a dynamic fuzzy OWA model to deal with problems of group multiple criteria decision making. The proposed model can help users to solve MCDM prob-lems under the situation of fuzzy or incomplete information. The advantages of this study are:
(1) The proposed approach can modify associated dynamical weights based on the aggregation situation (information capacity).
(2) The fuzzy linguistic variables are used to help the deci-sion maker to obtain the criteria weights more flexibly and reasonably (based on situation).
(3) The fuzzy OWA model can work like a “magnifying lens” to enlarge/find the most important attribute, which is dependent on the sparest information (i.e. optimistic case:
situation parameter α= 0 or 1), or obtain equal weights of attributes based on maximal information (i.e.moderate case: situation parameter α= 0.5).
Future studies may find that the applications of some other group MCDM problems. Or, the proposed model can be adapted to fuzzy neuron network (FNN) for achieving a bet-ter solution to uncertain problems. Fuzzy rule base of FNN can be combined with the fuzzy OWA model presented here, which can help accelerate the convergence of the fuzzy rule base.
References
1. Borcherding K, Epple T, Winterfeldt DV (1991) Comparison of weighting judgments in multi-attribute utility measurement. Man-age Sci 37(12):1603–1619
2. Chen SM (2001) Fuzzy group decision making for evaluating the rate of aggregative risk in software development. Fuzzy Sets Syst 118:75–88
3. Chen YC (2002) An application of fuzzy set theory to the exter-nal performance evaluation of distribution centers in logistics. Soft Comput 6:64–70
4. Choi DY (1999) A new aggregation method in a fuzzy environment. Decis Support Syst 25:39–51
5. Choo EU, Wedley WC (1985) Optimal criteria weights in repetitive multicriteria decision making. J Oper Res Soc 37:527–541 6. Darmon RY, Rouzies D (1991) Internal validity assessment of
con-joint estimated attribute importance weights. J Acad Marketing Sci 19(4):315–322
7. Dyer JS, Sarin RK (1979) Measurable multiattribute functions. Oper Res 27:810–822
8. Fuller R, Majlender P (2001) An analytic approach for obtaining maximal entropy OWA operator weights. Fuzzy Sets Syst 124:53– 57
9. Jaynes ET (1989) Cleaning up mysteries: the original goal, maxi-mum entropy and Bayesian methods. Kluwer, Dordrecht 10. Jaynes ET, Rosen krantz RD (1983) Papers on probability, statistics
and statistical physics. D. Reidel, Dordrecht
11. Klir GJ (1988) Fuzzy sets, uncertainly and information. Prentice Hall, Englewood Cliffs
12. Klir GJ, Wierman MJ (1999) Uncertainty-based information, 2nd edn. Physica-Verlag, Germany
13. Lee HM (1996) Group decision making using fuzzy sets theory for evaluating the rate of aggregative risk in software development. Fuzzy Sets Syst 80:261–271
14. Mendel JM (2000) Uncertain rule-based fuzzy logic systems: intro-duction and new directions. Prentice Hall PTR, Upper Saddle River 15. Mesiar R, Saminger S (2004) Domination of ordered weighted
averaging operators over t-norms. Soft Comput 8:562–570 16. Moshkovich HM, Schellenberger RE, Olson DL (1998) Data
influ-ences the result more than preferinflu-ences: some lessons from imple-mentation of multiattribute techniques in a real decision task. Decis Support Syst 22:73–84
17. Nutt PC (1980) Comparing methods for weighting decision crite-ria. OMEGA 8:163–172
18. O’Hagan M (1988) Aggregating template or rule antecedents in real-time expert systems with fuzzy set logic. In: Proceedings of 22nd Annual IEEE Asilomar conference on signals, systems, com-puters, Pacific Grove, pp 681–689
19. Pekelman D, Sen S (1974) Mathematical programming models for determination of attribute weights. Manage Sci 20(8):1217–1229 20. Ribeiro RA, Pereira RAM (2003) Generalized mixture operators
using weighting functions: a comparative study with WA and OWA. Eur J Oper Res 145:329–342
21. Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423
22. Shoemaker PJH, Carter WC (1982) An experimental comparison of different approaches to determining weights in additive utility models. Manage Sci 28:182–196
23. Smolikova R, Wachowiak MP (2002) Aggregation operators for selection problems. Fuzzy Sets Syst 131:23–34
24. Solymosi T, Dombi J (1986) A method for determining the weights of criteria: the centralized weights. Eur J Oper Res 26:35–41
25. Weber M, Eisenfhr F, von Winterfeldt D (1988) The effects of split-ting attributes on weights in multiattribute utility measurement. Manage Sci 34:431–445
26. Yager RR (1980) On a general class of fuzzy connectives. Fuzzy Sets Syst 4:235–242
27. Yager RR (1988) Ordered weighted averaging aggregation opera-tors in multi-criteria decision making. IEEE Trans Syst Man Cy-bern 18:183–190
28. Yager RR (1991) Connectives and quantifiers in fuzzy sets. Fuzzy Sets Syst 40:39–75
29. Yager RR, Kacprzyk J (1997) The ordered weighted averaging operators. Kluwer, Boston
30. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353
31. Zhang D, Yu PL, Wang PZ (1992) State-dependent weights in mul-ticriteria value functions. J Optimization Theory Appl 74(1):1–21 32. Zimmermann HJ, Zysno P (1980) Latent connectives in human