Chapter 2 Background and Literature Review – Towards new research approaches to enhance
3.8 The Proposed FCM Fuzzy Aggregation Process
According to the literature, obtaining a consensus FCM from a group of different FCMs pertaining to stakeholders with diverse perceptions is currently a challenge. The FCMs are typically developed for complex domains characterised by uncertain and imprecise knowledge. As such, the issue of conflicting perceptions between the stakeholders (designers of FCMs) is natural to arise. In real-life problems, the incorporation of the human dimension through human perceptions/opinions has become a necessity to fully characterise and study such problems. However, people vary in the level of knowledge and experience and their opinions should be taken accordingly when they are combined into an overall expression for a group. Hence, the FCMs that depict these perceptions should be weighted according to an objective credibility measure. This issue constitutes a barrier for developing a valid and accurate group FCM that describes a consensus perception. Another issue that has been stressed throughout this thesis is that different people may need different measures to express their knowledge in the form of nodes and connection between nodes. Some of them could use linguistic terms; others could use numeric values; or they may use different scales of linguistic and numeric measures. Finally, as complex problems addressed by FCM are typically characterised by ambiguity and uncertainty, the imprecise values describing the connections between nodes in FCM should be represented by a suitable model that retains the accuracy when combining weights expressed in different formats in different FCMs. This important issue and the other issues stated above make proposing an appropriate and robust FCM aggregation method a challenge.
Our FCM fuzzy aggregation method has been proposed to overcome this challenge. The FCM aggregation can be applied on the FCMs before or after condensation process, or even at any level of it. In this method, the imprecise connection values between nodes in FCMs are represented in a novel fuzzy way throughout the FCM aggregation process by using a robust fuzzy representation model. As described previously, for this fuzzy representation of the imprecise connection values, the proposed method uses the 2-tuple fuzzy linguistic representation model to represent them in fuzzy β values. The aggregation method also takes into consideration the importance of these connection values by weighting them according to the credibility weight of their FCMs before combining them with the connection values in other FCMs. The calculation of the credibility weight cwk for each FCMkwas also
92 linguistic and numeric scales that describe these imprecise connection values. In other words, the method allows describing the imprecise connections by different fuzzy sets as appropriate. Fuzzy sets could include different number of linguistic terms represented by different membership functions (fuzzy subsets) to deal with multiple linguistic and/or numeric scales. The different linguistic sets are to be converted into one uniform set called Base Linguistic Term Set (BLTS) (Herrera et al., 2000; Herrera & Martinez, 2000). The BLTS
is chosen such that it contains a number of Linguistic Terms appropriate to represent the overall group of FCMs being assembled. This process is called the normalization of the linguistic sets and will be described later on in this section. Algorithm 3.7 describes the proposed FCM fuzzy aggregation process of multiple FCMs (original or condensed) into a single FCM (group or social) and Appendix A.5 includes the detailed MATLAB code and the description of the calculations.
Algorithm 3.7. The steps of the fuzzy aggregation of multiple FCMs into a group or social FCM
1. Identify FCMs to be combined into a group or social FCM
2. Define a uniform linguistic term set as a Base Linguistic Term Set (BLTS) to deal
with all possible linguistic terms that represent the connection weight values of the identified FCMs
3. Count the number of distinct nodes in all identified FCMs (NC)
4. Initialize an adjacency matrix (Soc) for the group or social FCM and fill its
elements by zero values as follows:
0 i, j 1 NC}Socij (3.16)
Where Socij represents the connection weight between node ci and node cj in
the group or social FCM
5. For each FCM (FCMk) do the following:
5.1Recall the β connection weight values ofFCMk
5.2Recall the credibility weight cwkofFCMk
5.3Convert the βvalues of FCMk into the interval [-1, 1]
93
5.5Make the weight values of FCMk uniform with BLTS by converting these
values represented by membership functions assessed in S to values
represented by membership functions assessed in BLTS(Normalization
Process)
5.6Convert the resulting values into βvalues
5.7Augment its matrix to include all distinct nodes in all FCMs
5.8Fill the columns and rows of new nodes by zero weights
5.9Weight FCMkconnection weight values using its credibility weight cwk as
follows:
k k
wk FCM cw
FCM (3.17)
whereFCMwk is the resulting weighted matrix of FCM obtained from
multiplying the FCM connection weight valuesFCMk by its cwk
5.10Combine the weighted augmented matrix, FCMwk with the group/social
matrix,Soc, as follows:
wk
FCM
Soc
Soc
(3.18)6. Repeat step 5 for all identified FCMs
The FCM aggregation process given in Algorithm 3.7 can be briefly summarised as follows: The process starts by identifying: the FCMs to be aggregated into an FCM group, BLTS, and the number of distinct nodes in these FCMs. According to the number of distinct nodes, we initialize a zero adjacency matrix for the group FCM to include all possible nodes and connection weights between them that could result from combining the FCMs into the group. Then, for each FCMk in the group, we recall its connection values, cwk, number of nodes, and type of linguistic term set and number of terms in this set. Before combining any FCM, the proposed method converts the fuzzy subsets of FCM into fuzzy subsets in the
BLTS. The objective of this process is to manage the imprecise connection values of FCMs in a way that prevents any loss of information during the aggregation process by normalizing the fuzzy subsets representing these imprecise values into a standard set, BLTS (Herrera et al., 2000). To do so, we convert the connection values of FCMs into the interval [-1, 1];
94 the procedure for this step was explained previously. Then, we transform the linguistic set of FCM into the BLTS.
Let S (sp,,sp) be the linguistic set of an FCM and BLTS (tg,,tg) be the linguistic set of the BLTS, such that p g , then the function TSBLTS defines each fuzzy subset of Sin the BLTS as follows (Herrera et al., 2000)
S
s
g
k
t
s
T
SBLTS(
i){(
k,
ki),
{0,,
}},
i
(3.19))})
(
),
(
(min{
max
y
y
k i t s y i k
(3.20)where maxandminare the usual maximum and minimum operations,respectively, yis a (connection) value on [-1, 1] universe of discourse, and (y)
i s and (y) k t are the membership functions of the fuzzy subsets associated with the terms si and tk, respectively.
For further explanation about this normalization process, please see (Herrera et al., 2000). Appendix A.5 also describes its calculations. As the next step in the process, using the transformed uniform linguistic set of FCM, a conversion of its connection values in the interval [-1, 1] back to β values is performed. This step is followed by an FCM augmentation process; here, the adjacency matrix of each FCMk is augmented to include all nodes in all FCMs. The column and row of each new node added to the matrix are filled with zero values. The augmented matrix is then multiplied by the FCM credibility weight cwk(Equation 3.17). Finally, the weighted augmented matrix is aggregated with the adjacency matrix of group FCM (Equation 3.18). The above steps of the FCM aggregation process are repeated until the last FCM in the group has been added to the group FCM.
To aggregate the example ‘expert’ and ‘local people’ FCMs presented in Figure 2.13 in Chapter 2 into a group FCM, consider their β values in Table 2.4 in Chapter 2. Suppose the results of calculating the credibility weights of the ‘expert’ and ‘local people’ FCMs are 0.6
and 0.4, respectively, and suppose the linguistic term set S used to represent their connection weights is BLTSitself. Therefore, there is no need for the normalization process. Applying the steps of the fuzzy aggregation in Algorithm 3.7, the β values of the resulting group FCM are presented in Table 3.7. This table shows that the group FCM
95 includes all distinct nodes and all possible connections between these nodes that existed in both FCMs. Figure 3.4 shows ‘expert’ and ‘local people’ and resulting group FCMs. It shows the connection weights of these FCMs are represented by 2-tuple values.
Table 3.7 The β values of the group FCM resulting from the fuzzy aggregation of the ‘expert’ and
‘local people’ FCMs presented in Chapter 2
Water Situation Water Resources Water Demand Technology Wastage of Water Water Projects Economic Situation Water Situation 0 0 0 0 0 0 1.200 Water Resources 5.247 0 0 0 0 0 0 Water Demand -6.000 0 0 0 0 0 0 Technology 0 2.153 -1.088 0 -2.513 0 0 Wastage of Water -1.447 0 0.713 0 0 0 0 Water Projects 0 1.600 0 0 0 0 -2.000 Economic Situation 0 0 -1.200 0 0 1.600 0
Figure 3.4 A graphical representation of aggregating ‘expert’ and ‘local people’ FCMs presented in Chapter 2. The connection weights of all FCMs are represented by 2-tuple values
In addition to individual (original) FCMs, the condensation and aggregation processes create a number of diverse FCMs (group FCMs and a social group FCM) to be analysed and simulated. This allows us to reach towards a comprehensive knowledge and understanding of the problem and allows access to a wide range of individual and group perceptions, such
96 that analysing and simulating these perceptions lead to many reliable suggestions or potential solutions to solve the problem.