This document discusses a new technique called Delphi Adapted Fuzzy Associative Memories (DAFAM) that functions as a multiple expert system to combine the views of different experts. It explains how DAFAM works and applies it to investigate the impacts of climate change on the environment based on the opinions of four experts. The antecedent attributes related to causes of climate change are combined with consequent attributes of its effects using DAFAM to generate a single fuzzy relation matrix representing the collective views of the experts.

1. Integrated Intelligent Research(IIR) International Journal of Business Intelligent
Volume: 04 Issue: 01 June 2015,Pages No.30- 33
ISSN: 2278-2400
30
Delphi Adapted Fuzzy Associative Memories
(DAFAM) as a Multiple Expert System and its
Application to Study the Impacts of Climate
Change on Environment
A.Victor Devadoss1
, D.Ajay2
, K.Sudha3
1
Head & Associate Professor, Department of Mathematics, Loyola College, Chennai-34.
2,3
Ph.D Research Scholar, Department of Mathematics, Loyola College, Chennai-34.
Email: hanivictor@ymail.com, dajaypravin@gmail.com, ashu.8788@gmail.com
Abstract-A new fuzzy technique Delphi Adapted FAM is
proposed in this paper and is used to investigate the impacts of
climate change on environment. DAFAM functions as a
multiple expert system, in that it can be used to combine any
number of expert’s views into one relational matrix. The first
section of this paper gives an introduction to what is done in the
paper and the second section explains the dynamics of FAM.
Section three explains the technique DAFAM and in the fourth
section DAFAM is adapted to investigate the impacts of climate
change on the environment. Section five, the last section derives
the conclusion and makes some suggestions.
Keywords: Delphi adapted Fuzzy models, Multiple expert
system, Fuzzy associative memories, Climate change.
I. INTRODUCTION
The Delphi method (Dalkey & Helmer, 1963) is a proven tool
for collective decision making (Linstone & Turoff, 2002) for a
situation in which decision needs to be made by a group of
experts who might have divergent views on the topic. This
method can also be called a prediction method based ob expert
judgment. The Delphi method is characterized by the
properties like anonymity, feedback, statistical and
convergence. It tries to achieve a consensus among the experts.
In order to account for the amount of fuzziness in group
decision making Murray, Pipino & Gigch (1985) proposed
Fuzzy Delphi method. Since then the method has found many
applications. Fuzzy associative memories (FAM) as a model
has been applied to analyse problems in which the factors that
attribute to the problem can be classified into antecedent and
consequent sets and the relationship between them needs to be
analysed. In this paper, the novelty of Fuzzy Delphi Method in
bringing a consensus is combined with Fuzzy associative
memories so that the new technique thus obtained can function
as a multiple expert system.
II. FUZZY ASSOCIATIVE MEMORIES (FAM)
A fuzzy set is a map μ: X → [0, 1] where X is any set called
the domain and [0, 1] the range. That is to every element x X,
μ assigns membership value in the interval [0, 1]. Fuzzy
theorists often picture membership functions as two-
dimensional graphs with the domain X represented as a one-
dimensional axis.The geometry of fuzzy sets involves both
domain 1 2
( , ,... )
n
X x x x
 and the range [0, 1] of mappings μ:
X → [0, 1]. A fuzzy subset equals the unit hyper cube
[0,1]
n n
I  . The fuzzy set is a point in the cube
n
I . Vertices
of the cube
n
I define a non-fuzzy set. Now within the unit
hyper cube [0,1]
n n
I  we are interested in distance between
points, which led to measures of size and fuzziness of a fuzzy
set and more fundamentally to a measure. Thus within cube
theory directly extends to the continuous case when the space
X is a subset of
n
R . The next step is to consider mappings
between fuzzy cubes. A fuzzy set defines a point in a cube. A
fuzzy system defines a mapping between cubes. A fuzzy
system S maps fuzzy sets to fuzzy sets. Thus a fuzzy system S
is a transformation : n p
S I I
 . The n-dimensional unit
hyper cube In
houses all the fuzzy subsets of the domain space
or input universe of discourse 1 2
( , ,... )
n
X x x x
 .
p
I houses
all the fuzzy subsets of the range space or output universe of
discourse, 1 2
( , ,..., )
p
Y y y y
 . X and Y can also denote
subsets of
n
R and
p
R . Then the fuzzy power sets (2 )
X
F
and (2 )
Y
F replace
n
I and
p
I .In general a fuzzy system S
maps families of fuzzy sets to families of fuzzy sets thus
1 1
: ... ... s
r p
n n p
S I I I I
     . Here too we can extend
the definition of a fuzzy system to allow arbitrary products or
arbitrary mathematical spaces to serve as the domain or range
spaces of the fuzzy sets. We shall focus on fuzzy systems
: n p
S I I
 that map balls of fuzzy sets in
n
I to balls of
fuzzy set in
p
I . These continuous fuzzy systems behave as
associative memories. The map close inputs to close outputs.
We shall refer to them as Fuzzy Associative Maps or
FAMs.The simplest FAM encodes the FAM rule or association

2. Integrated Intelligent Research(IIR) International Journal of Business Intelligent
Volume: 04 Issue: 01 June 2015,Pages No.30- 33
ISSN: 2278-2400
31
( , )
i i
A B , which associates the p-dimensional fuzzy set i
B
with the n-dimensional fuzzy set i
A . These minimal FAMs
essentially map one ball in
n
I to one ball in
p
I . They are
comparable to simple neural networks. But we need not
adaptively train the minimal FAMs. In general a FAM system
: n b
F I I
 encodes the processes in parallel a FAM bank of
m FAM rules 1 1
( , )...( , )
m m
A B A B . Each input A to the FAM
system activates each stored FAM rule to different degree.The
minimal FAM that stores ( , )
i i
A B maps input A to
'
i
B a
partly activated version of i
B . The more A resembles i
A , the
more
'
i
B resembles i
B . The corresponding output fuzzy set B
combines these partially activated fuzzy sets
1 1 1
1 2
, ,..., m
B B B .
B equals a weighted average of the partially activated sets
1 1
1 1 ... n m
B w B w B
   where i
w reflects the credibility
frequency or strength of fuzzy association ( , )
i i
A B . In
practice we usually defuzzify the output waveform B to a
single numerical value j
y in Y by computing the fuzzy
centroid of B with respect to the output universe of discourse
Y .More generally a FAM system encodes a bank of
compound FAM rules that associate multiple output or
consequent fuzzy sets
1
,..., s
i i
B B with multiple input or
antecedent fuzzy sets
1
,..., r
i i
A A . We can treat compound
FAM rules as compound linguistic conditionals. This allows us
to naturally and in many cases easily to obtain structural
knowledge. We combine antecedent and consequent sets with
logical conjunction, disjunction or negation. For instance, we
could interpret the compound association
1 2
( , , )
A A B ;
linguistically as the compound conditional “IF
1
X is
1
A
AND
2
X is
2
A , THEN Y is B ” if the comma is the fuzzy
association
1 2
( , , )
A A B denotes conjunction instead of say
disjunction.We specify in advance the numerical universe of
discourse for fuzzy variables
1 2
,
X X and Y . For each
universe of discourse or fuzzy variable X, we specify an
appropriate library of fuzzy set values
2
1 ,...
r
k
A A Contiguous
fuzzy sets in a library overlap. In principle a neural network
can estimate these libraries of fuzzy sets. In practice this is
usually unnecessary. The library sets represent a weighted
though overlapping quantization of the input space X. They
represent the fuzzy set values assumed by a fuzzy variable. A
different library of fuzzy sets similarly quantizes the output
space Y. Once we define the library of fuzzy sets we construct
the FAM by choosing appropriate combinations of input and
output fuzzy sets Adaptive techniques can make, assist or
modify these choices.
III. DELPHI ADAPTED FAM (DAFAM)
Consider a system of k synaptic connection matrices
     
(1) (2) ( )
1 2
, ,....... k
ij ij k ij
N b N b N b
   where
1,2,....,
i n
 and 1,2,....,
j p
 which represent opinions of
k experts about causal relationship between the neuron field
x
F with nneurons and the neuron field y
F with p neurons.
Then these matrices are combined as one synaptic connection
matrix  
ij
M a
 where
, 1,2,..., 1,2,..., ........(1)
3
ij ij ij
ij
P M O
a i n and j p
 
    Here
( )
1
min{ }
n
ij ij
n k
P a
 
 ,
( )
1
1 k
n
ij ij
n
M a
k 
  , and
( )
1
max{ }
n
ij ij
n k
O a
 

, where 1,2,...., , 1,2,....., , 1,2,....,
i n j p n k
   and
( )
1,2,...,
n
ij n
a M n k
   .
Fig.1 DAFAM as a multiple expert system
𝑁1 = (𝑏𝑖𝑗
(1)
)
𝑁2 = (𝑏𝑖𝑗
(2)
)
𝑁𝑘 = (𝑏𝑖𝑗
(𝑘)
)
Expert k
Expert 2
Expert 1
𝑀 = (𝑎𝑖𝑗)
𝑋𝑖𝑀 = 𝑋𝑖
′
𝑋𝑖
′
𝑀𝑡
= 𝑌𝑖
Defuzzification

3. Integrated Intelligent Research(IIR) International Journal of Business Intelligent
Volume: 04 Issue: 01 June 2015,Pages No.30- 33
ISSN: 2278-2400
32
IV. ADAPTATION OF DAFAM TO THE PROBLEM
Let us consider that there are n attributes, say x1, … , xn, where
n is finite, that are associated with the causes of climate change
and let y1, … ,yp be the attributes associated with its effects,
where p is finite. We choose the following attributes, which are
causes of climate change, as nodes in the domain space of
FAM.
C1 – Green house gas (GHG) emissions
C2 – Natural factors
C3 – Solid waste management
C4 – Deforestation
C5 – Water and land pollution
The following are the attributes we choose as effects of climate
change.
G1 – Migration of species
G2 – Crop yields
G3 – Glacier and snowpack decline
G4 – Sea level rise
G5 –Species extinction
G6 –Rise in average temperature
G7 –Spread of new diseases
The fuzzy relation between these antecedent and consequent
sets of the problem as given by four different experts is given
by the following matrices:
1
.8 .7 .9 .9 .8 1 .5
.6 .7 .3 .4 .2 .4 .3
.6 .8 .6 .3 .1 .3 .7
.9 .8 .6 .5 .8 .8 .3
1 .9 .8 .8 .6 .7 .9
M
 
 
 
 

 
 
 
 
2
.6 .7 .8 .5 .8 1 .3
.5 .3 .5 .4 0 .2 0
.4 .9 .6 .7 0 0 .8
.7 .9 .8 .3 .7 .8 0
.5 .4 0 0 .6 .2 .9
M
 
 
 
 

 
 
 
 
3
.8 .7 .7 .6 .8 1 .3
.5 .2 .5 .4 0 0 .2
.5 .9 .6 .7 .4 0 .8
.8 .5 .8 .3 .7 .8 0
.5 .7 0 .2 .6 0 .8
M
 
 
 
 

 
 
 
 
4
.8 .5 .7 .6 .8 1 .5
0 .2 .5 .4 0 0 .2
.6 .8 .5 .2 .4 0 .8
.8 .6 .5 0 .7 .8 .5
.7 .7 .2 .8 .6 .2 .8
M
 
 
 
 

 
 
 
 
The procedure to combine the views of four experts using equation (1) of DABAM is shown in the table below:
G1 G2 G3 G4 G5 G6 G7
P M O P M O P M O P M O P M O P M O P M O
C
1
0.6
0.75
0.
8
0.
5
0.65
0.
7
0.
7
0.77
5
0.
9
0.
5
0.65
0.
9
0.
8
0.8 0.8 1 1 1
0.
3
0.4
0.
5
C
2
0 0.4
0.
6
0.
2
0.35
0.
7
0.
3
0.45
0.
5
0.
4
0.4
0.
4
0 0.05 0.2 0 0.15
0.
4
0
0.17
5
0.
3
C
3
0.
4
0.52
5
0.
6
0.
8
0.85
0.
9
0.
5
0.57
5
0.
6
0.
2
0.47
5
0.
7
0
0.22
5
0.4
0
0.07
5
0.
3
0.
7
0.77
5
0.
8
C
4
0.
7
0.8
0.
9
0.
5
0.7
0.
9
0.
5
0.67
5
0.
8
0
0.27
5
0.
5
0.
7
0.72
5
0.8
0.
8
0.8
0.
8
0 0.2
0.
5
C
5
0.5 0.67
5
1
0.
4
0.67
5
0.
9
0 0.25
0.
8
0 0.45
0.
8
0.
6
0.6 0.6 0
0.27
5
0.
7
0.
8
0.85
0.
9

4. Integrated Intelligent Research(IIR) International Journal of Business Intelligent
Volume: 04 Issue: 01 June 2015,Pages No.30- 33
ISSN: 2278-2400
33
The related combined fuzzy matrix M formulated using the
opinions of the experts is as follows:
0.717 0.617 0.792 0.683 0.8 1 0.4
0.333 0.417 0.417 0.4 0.083 0.183 0.158
0.508 0.85 0.558 0.458 0.208 0.125 0.758
0.792 0.7 0.658 0.258 0.742 0.8 0.233
0.725 0.658 0.35 0.417 0.6 0.325 0.85
M
 
 
 
 

 
 
 
 
Consider a fit vector C1 = (1 0 0 0 0) where the node Green
house gas emission is kept in ON state.
C1M = (0.717 0.617 0.792 0.683 0.8 1 0.4)
↪ (0 0 0 0 1 1 0) = 𝐶1
′
𝐶1
′
𝑀𝑇
= (1.8 0.266 0.333 1.542 0.925)
↪ (0 0 0 1 0) = X1
X1M = (1.509 1.317 1.45 0.941 1.542 1.8 0.633)
↪ (0 0 0 0 1 1 0) = 𝑋1
′
𝑋1
′
𝑀𝑇
= (1.8 0.266 0.333 1.542 0.925)
↪ (1 0 0 1 0) = X2 = X1
Therefore the binary pair (0 0 0 0 1 1 0), (1 0 0 1 0)
represents the fixed point.
The following table gives different limit points when we take
different input vectors
Input vector Limit cycle
(1 0 0 0 0) (0 0 0 0 1 1 0), (1 0 0 1 0)
(0 1 0 0 0) (1 1 0 0 0 0 0), (0 0 0 1 1)
(0 0 1 0 0) (0 1 0 0 0 0 1), (0 0 1 0 1)
(0 0 0 1 0) (0 0 0 0 1 1 0), (1 0 0 1 0)
(0 0 0 0 1) (0 1 0 0 0 0 1), (0 0 1 0 1)
V. CONCLUSION AND SUGGESTIONS
It can be inferred from the table above that the Delphi adapted
FAM brings out the factor which affects the most or gets
affected the most in the situation under analysis. It should be
noted here that the results derived here are the combined
opinion of different experts. When C1 is kept in ON state C4
also turns into ON state and they have a combined effect on G5
and G6. That is, GHG emission and deforestation have great
impact on species extinction and species extinction and result
in an increase in average temperature. Similarly other results
can also be interpreted. The limit points obtained through
DABAM suggests that C5 (water and land pollution) is the
most important factor that affects the environment. Therefore
pollution should be controlled to protect the environment from
being affected by climate change. It can also be inferred from
the table that C1 (Green house gas (GHG) emissions), C3 (Solid
waste management) and C4 (Deforestation) are the next
important factors that contribute to the degradation of the
environment. Finally the combined opinion of the experts is
that C2 (Natural factors) does not play crucial role in affecting
the environment when compare to the other factors.
REFERENCES
[1] Dalkey, N., & Helmer, O. (1963). An experimental application of the
Delphi method to the use of experts. Management Science, 9, 458–467.
[2] Devadoss, A.V., et al (2014). Analysing the impacts of Climate change
using Fuzzy associative memories model (FAM), International Journal
of Computing Algorithm, Volume: 02, October 2013, Pages:373-377.
[3] “Climate change 2007: synthesis report” a report by IPCC.
[4] Kosko B., Neural Networks and Fuzzy Systems, Prentice-Hall, Inc.,
New Jersey, USA, 1992.
[5] Linstone, H. A., & Turoff, M., ed, 2002, The Delphi Method:
Techniques and Applications, ISBN 0-201-04294-0.
[6] McMichael. A.J., et al (Ed)., Climate and Human Health: Risks and
Responses., WHO, Geneva, 2003.
[7] Murray, T. J., Pipino, L. L., & Gigch, J. P. (1985). A pilot study of
fuzzy set modification of Delphi. Human Systems Management, 6–80.