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Genetic algorithms for decision support in the case of Fuzzy cognitive maps.


Axelord (1976) proposed cognitive maps as a formal tool for decision- making, using the matrix representation of the directed graph to represent and study the social scientific knowledge. Based on the cognitive map Cognitive maps, mental maps, mind maps, cognitive models, or mental models are a type of mental processing (cognition) composed of a series of psychological transformations by which an individual can acquire, code, store, recall, and decode information about the relative locations  structure Kosko (1986) proposed Fuzzy Cognitive Maps. Kosko(1988) had applied FCM FCM

See: Futures commission merchant


See futures commission merchant (FCM).
 to neural network neural network or neural computing, computer architecture modeled upon the human brain's interconnected system of neurons. Neural networks imitate the brain's ability to sort out patterns and learn from trial and error, discerning and extracting  and studied the hidden pattern in combined and adaptive knowledge networks. Cognitive Maps were used by Craiger and Coovert 1994) to study the public health scenario of a society, where the edges of the map were assigned a positive or a negative value. Craiger et al(1996) studied modeling organization behaviour with FCM. They had studied FCM as a computer simulation methodology for representing and predicting the behaviour of models of arbitrary complexity. Chong et al (2004) have used FCM in goal oriented decision support systems. A fuzzy cognitive map A Fuzzy cognitive map is a cognitive map within which the relations between the elements (e.g. concepts, events, project resources) of a "mental landscape" can be used to compute the "strength of impact" of these elements.  is a directed graph with feedback. FCM nodes represent concepts like policies, events, inputs, outputs, trends, goals etc. and edges represent causal links among the concepts. Each concept variable is given an initial value based on the belief of the expert(s) about the current state. The FCM is then free to interact until an equilibrium is reached. This is arrived at through a process of forward chaining. Given an initial state of a system represented by a set of values of its constituent concepts, an FCM can stimulate its evolution over time to predict its future behavior.

We may start with a goal and aim to identify the initial state that can trigger a chain of events leading to the state corresponding to the goal. The nonlinear transformation involved in computing successive FCM states and the process of reversing the matrix multiplication Noun 1. matrix multiplication - the multiplication of matrices
matrix operation - a mathematical operation involving matrices
 make the backward chaining In AI, a form of reasoning that starts with the conclusion and works backward. The goal is broken into many subgoals or sub-subgoals which can be solved more easily. Known as top-down approach. Contrast with forward chaining.  difficult. Hence in this paper we use a methodology based on the use of FCM and Genetic Algorithm genetic algorithm - (GA) An evolutionary algorithm which generates each individual from some encoded form known as a "chromosome" or "genome". Chromosomes are combined or mutated to breed new individuals.  for the goal-oriented procedure of a given scenario starting with the desired goal. We apply this procedure for sample study of cancer patients taking treatment in Thanjavur Medical College, Thanjavur, Tamil Nadu Tamil Nadu (tăm`əl nä`d), formerly Madras (mədrăs`, mədräs`), state (2001 provisional pop. . Most of the patients were agricultural labourers. Smoking tobacco, chewing tobacco chewing tobacco,
n See smokeless tobacco.

chewing tobacco Smokeless tobacco, see there
 and using alcohol are common habits among them. We analyse the factors of tobacco and alcohol related cancers using the mentioned methodology.


FCM graph consists of nodes (concepts, agents) and weighted arcs (connection, edge). Nodes on the graph represent concepts describing behavioural characteristics of the system. Concepts can be inputs, outputs, variables, states, events, actions and goals of the system. Signed weighted arcs represent causal relationships that exist among concepts. The edge weight Eij represents the degree of causal relationship between concept i and concept j. This setup gives rise to the following three types of interactions:

[E.sub.ij] > 0, a positive causality, where an increase in the value of the ith concept causes an increase in the value of the jth concept. [E.sub.ij] < 0, a negative causality, where an increase in the value of the ith concept causes a decrease in the value of the jth concept. [E.sub.ij] = 0, no causal relationship between the ith concept and jth concept.

Kosko (1992) proposed a rule to calculate the value of each concept based on the influence of the interconnected concepts which involves vector-matrix multiplication and the forward chaining involves a series of repeated vector-matrix multiplication and transformation. At each time step t, the state vector
  • A quantum state vector fully specifies any quantum mechanical state in which a quantum mechanical system can be.
  • A geographical state vector specifies the position and velocity of an object in space.
 is calculated as

[C.sub.t] = F([C.sub.t-1] x E)

where [C.sub.t] is the state vector at time step t, [E.sub.ij] is the weight of the arc connecting [C.sub.i] to [C.sub.j] and E is the weight matrix and F(X) = ([f.sub.1], [f.sub.2], ..., [f.sub.n]) is a thresholding function operating on the resulting vector. Generally, a sigmoid function “S-curve” redirects here. For the recording company, see S-Curve Records.

A sigmoid function is a mathematical function that produces a sigmoid curve — a curve having an "S" shape.
 f(x) = 1/(1 + [e.sup.-[lambda]x]) is used for thresholding to constrain the value of f (x) in the interval [0, 1]. After the implementation of thresholding we update the resulting vector by clamping the concepts of interest. This new output vector is treated as the input vector for the next stage.

The vector matrix multiplication operation to derive successive FCM states is iterated until the FCM converges or settles down to one of the following:

(i) A fixed point attractor Point Attractor

In non-linear dynamics, an attractor where all orbits in phase space are drawn to one point, or value. Essentially, any system which tends to a stable, single valued equilibrium will have a point attractor.
: The FCM state vector remains unchanged for successive iterations. This stable state is known as the fixed point attractor.

(ii) A Limit Cycle: A sequence of FCM state keeps repeating indefinitely. This sequence is known as limit cycle.

(iii) A chaotic attractor Noun 1. chaotic attractor - an attractor for which the approach to its final point in phase space is chaotic
strange attractor

attracter, attractor - (physics) a point in the ideal multidimensional phase space that is used to describe a system toward which
: The FCM state vector keeps changing. Repeating states are never found.


Genetic Algorithms (GA) were first introduced by John Holland (1975) . GA attempts to arrive at optimal solutions through a process similar to biological evolution. This involves following the principles of survival of the fittest, crossbreeding and mutation to generate better solutions from a pool of existing solutions. Genetic algorithms have been found to be capable of finding solutions for a wide variety of problems for which no acceptable algorithmic solutions exist. The GA methodology is particularly used for optimization, a problem solving problem solving

Process involved in finding a solution to a problem. Many animals routinely solve problems of locomotion, food finding, and shelter through trial and error.
 technique in which one or more very good solutions are searched for in a solution space consisting of a large number of possible solutions. GA reduce the search space by continually evaluating the current generation of candidate solutions. It discards the solutions ranked as poor, and produces a new generation through crossbreeding and mutating those ranked as good. The ranking of candidate solutions is done using some pre-determined measure of fitness.

The GA evolutionary cycle starts with a randomly selected initial population. The changes to the population occur through the processes of selection based on fitness, and alteration using crossover and mutation. The application of selection and alteration leads to a population with a higher proportion of better solutions. The evolutionary cycle continues until an acceptable solution is found in the current population, or some control parameter such as the number of generations is exceeded.


The steps in the typical genetic algorithm for finding a solution to a problem are listed below:

1. Create an initial solution population of a certain size randomly.

2. Evaluate each solution in the current generation and assign it a fitness value.

3. Select "good" solutions based on fitness value and discard the rest.

4. If acceptable solution(s) are found in the current generation, or maximum number of generations is exceeded, then stop.

5. Alter the solution population using crossover and mutation to create a new generation of solutions.

6. Go to step 2.

The study reported in this paper is to analyse the factors related to tobacco and alcohol related cancers by attempting to optimize the search for the initial starting state vector that will lead to a desired goal state.


Here our aim is to identify the initial vector that can cause an FCM to converge to the desired goal vector. Generally we use repeated vector-matrix multiplication when the initial vector is given. Conversely when the desired goal vector is taken as attractor and if we are to find the initial vector avoiding the process involving repeated matrix inverse etc. we use GA based methodology.

i. Start with the desired goal vector G = ([g.sub.1], [g.sub.2]........., [g.sub.n]) which we want to be the attractor.

ii. Define the policy vector P = ([p.sub.1], [p.sub.2].........., [p.sub.n]). This vector specifies for each node, whether it can be a policy node. If the ith node is allowed to be a policy node then pi=1. Otherwise pi is set to 0. When using the FCM we clamp the state vectors as per the information provided by the nodes of the respective policy vector.

iii. When we apply GA, we optimize the following objective function dA (A, G) = min (d) where d [epsilon] D, D = {d (a, G)/a[epsilon]A} and d (a, G) = Euclidean distance of a and G, a is the attractor the FCM converges to, after being fed with an initial vector. A is the set of all attractors, dA defines dissimilarity between the goal vector G and the attractor.

Let P(t) represent the binary coded population generated at time t using the generated initial vectors and policy vectors, attractors are found by applying FCM. Then objective function dA is evaluated. It is minimized by executing the steps based on Goldberg's simple GA as given below.
Set mutation rate to a predetermined rate
Set population size to a predetermined size
Initialize P(t) by generating a random population of initial starting
Evaluate P(t) using (iii)
While not finished do
Select P(t) from P(t-1)
Rank P(t) according to fitness
Remove the least fit 50% of P(t)
Replace those removed by the process of cross over and mutation of the
best fit 50% of
Evaluate P(t) by using (iii)
End while

As a result of above we are left with initial vectors and corresponding policy vectors. There may be more than one possible vector that lead to the state close to the Goal vector. GA is capable of finding many solutions simultaneously.


Tobacco addiction ranks as a greatest public health catastrophe of our time. The practice of inhaling cigarette smoke which gained widespread acceptance only during the twentieth century has generated devastating cancer outcomes for our society. Specially lung cancer lung cancer, cancer that originates in the tissues of the lungs. Lung cancer is the leading cause of cancer death in the United States in both men and women. Like other cancers, lung cancer occurs after repeated insults to the genetic material of the cell. , previously rare has risen to become the leading cancer killer. A synergistic, multiplicative effect appears to exist between smoking and drinking. Smoking is accepted as the major cause of Cancers of larynx larynx (lâr`ĭngks), organ of voice in mammals. Commonly known as the voice box, the larynx is a tubular chamber about 2 in. (5 cm) high, consisting of walls of cartilage bound by ligaments and membranes, and moved by muscles. , Pharynx pharynx (fâr`ĭngks), area of the gastrointestinal and respiratory tracts which lies between the mouth and the esophagus. In humans, the pharynx is a cone-shaped tube about 4 1-2 in. (11.43 cm) long. , oral cavity, esophagus and lungs and it is a contributory factor in cancers of the pancreas, bladder, kidney, colen and stomach. Spit Tobacco spit tobacco,
n See smokeless tobacco.
 in its various forms, loose-leaf tobacco, and twist tobacco causes cancers of the oral cavity. ETS ETS Educational Testing Service (nonprofit private educational testing and measurement organization)
ETS Emergency Telecommunications Service
ETS Electronic Trading System
ETS Engineering (&) Technical Services
 (Environmental Tobacco Smoke) also called second hand smoke (combination of smoke released from a burning cigarette between puffs and the smoke exhaled by the smoker) has double the amount of nicotine than main stream smoke and has a higher concentration of carcinogens. People who smoke are likely to drink alcohol and risk of developing cancer is high in people who use tobacco and alcohol. Unemployment, poverty and frustration induce people to those habits. The use of tobacco and alcohol is a social and psychological problem. Agricultural coolies, scavengers, workers are affected by those habits. There is no definite reason for using tobacco and alcohol and we are at the outset justified in using FCM to analyze the problem.Taking in to account of all this factors the nodes considered by an expert are given below.
[C.sub.1]    --   Parental use
[C.sub.2]    --   Less effective parents
[C.sub.3]    --   Smoking tobacco
[C.sub.4]    --   Chewing tobacco
[C.sub.5]    --   ETS (Environmental tobacco smoke)
[C.sub.6]    --   Using alcohol
[C.sub.7]    --   Frustration and Low self image
[C.sub.8]    --   Mouth cancer
[C.sub.9]    --   Throat Cancer
[C.sub.10]   --   Lung Cancer
[C.sub.11]   --   Stomach Cancer
[C.sub.12]   --   Malnutrition
[C.sub.13]   --   Unemployment and Poverty

The study is based on an FCM with 13 nodes. The causal weights suggested by the expert are as follows.
[C.sub.1][C.sub.3] = 0.6     [C.sub.1][C.sub.5] = 0.7

[C.sub.3][C.sub.5] = 0.7     [C.sub.3][C.sub.6] = 0.3

[C.sub.3][C.sub.11] = 0.3    [C.sub.3][C.sub.12] = 0.6

[C.sub.5][C.sub.9] = 0.6     [C.sub.5][C.sub.10] = 0.6

[C.sub.6][C.sub.12] = 0.7    [C.sub.7][C.sub.3] = 0.6

[C.sub.13][C.sub.12] = 0.3

[C.sub.1][C.sub.6] = 0.7     [C.sub.2][C.sub.3] = 0.5

[C.sub.3][C.sub.8] = 0.4     [C.sub.3][C.sub.9] = 0.6

[C.sub.4][C.sub.8] = 0.7     [C.sub.4][C.sub.9] = 0.3

[C.sub.6][C.sub.8] = 0.2     [C.sub.6][C.sub.9] = 0.2

[C.sub.7][C.sub.6] = 0.6     [C.sub.12][C.sub.11] = 0.6

[C.sub.2][C.sub.6] = 0.5

[C.sub.3][C.sub.10] = 0.8

[C.sub.4][C.sub.11] = 0.1

[C.sub.6][C.sub.11] = 0.8

[C.sub.13][C.sub.7] = 0.7

The goal of this experiment is to use the workings of the FCM backward chaining technique described in section 4. This experiment models the causal relationships between some life style factors/habits and certain types of cancers. The process of finding the appropriate stimulus vector allows one to investigate the possible causes of a particular type of cancer say throat cancer.

The problem under study may be summarized by the question: "What scenario can lead to throat cancer (represented by node [C.sub.9])"?

(i) The goal vector G is therefore set to (0 0 0 0 0 0 0 0 1 0 0 0 0) where the ninth element represents the maximum state value 1.0 for the concept throat cancer. All of the nodes, except throat cancer, mouth cancer, lung cancer and stomach cancer are potential policy nodes. Accordingly, the policy vector is set as (1 1 1 1 1 1 1 0 0 0 0 1 1).

(ii) The population size is set as 20 and we consider 200 iterations i.e., 200 population sets.

(iii) The stimulus vector generated by genetic algorithm is found to be (.76, .74, .99, .76, 0, 0, .45, .47, .46, .38, .1, .11, .46) with corresponding policy vector (1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0) which leads to a fixed point attractor (.76, .5, .99, .76, .77, .8, .58, .74, .8, .77, .74, .11, .5).


Thus the level of getting throat cancer is maximum ([C.sub.9] = .8) when the level of smoking tobacco is high ([C.sub.3] = .99). The levels of chewing tobacco ([C.sub.4] = .76) and environment tobacco smoke ([C.sub.5] = .77) being more or less equal in the attractor, it is inferred that both these factors contribute to the incidence of throat cancer in an equal degree. Thus the technique of FCM backward inferencing with Genetic algorithm support is much helpful for the sample study of throat cancer. This procedure can also be extended to other types of tobacco and alcohol related cancers.


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(2.) Chong, A., Shamim Khan, M. and Sebastian Khor.2004., "Fuzzy Cognitive Maps with Genetic Algorithm for Goal-oriented decision support", Intl. J. Uncer. Fuzzi. Knowledgebased Sys., World Scientific Publishing Established in 1981, World Scientific Publishing Company (WSPC) is one of the leading scientific publishers in the world, and the largest international scientific publisher in the Asia-Pacific region.  Company. (2004).

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(5.) George, J. Kleir. and Yuan, B.O., 1995.Fuzzy Sets and Fuzzy Logic fuzzy logic, a multivalued (as opposed to binary) logic developed to deal with imprecise or vague data. Classical logic holds that everything can be expressed in binary terms: 0 or 1, black or white, yes or no; in terms of Boolean algebra, everything is in one set or , Theory and Applications, Prentice Hall Prentice Hall is a leading educational publisher. It is an imprint of Pearson Education, Inc., based in Upper Saddle River, New Jersey, USA. Prentice Hall publishes print and digital content for the 6-12 and higher education market. History
In 1913, law professor Dr.
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R. Nallasamy *, K. Nirmala and K. Kalaiselvi, **

Department of Mathematics, K.N. Govt. Arts College for Women, Thanjavur (T.N.), India

* Department of Mathematics, National Institute of Technology, Tiruchirapalli (T.N.), India

** Additional Director of Medical Education, Directorate of Medical Education, Kilpauk, Chennai (T. N.) India
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Author:Nallasamy, R.; Nirmala, K.; Kalaiselvi, K.
Publication:Bio Science Research Bulletin -Biological Sciences
Date:Jul 1, 2006
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