**3. Fuzzification of** *X***,** *Y* **and** *Z* **in the creation of code vectors**

Before studying the technique of making the self/non-self discrimination to state if the patient can be operated or not we should first be able to compare different strings *v* = (*x* = *age*, *y = CRP*, *z = body weight*), *xX*, *yY*, *zZ*, to decide their grades of affinity (coverage). We thus should design sets of codes for each biological parameter.

The markers *age*, *CRP*, and *body weight* are measurable features. Hence, we intend to determine the collections of codes assisting intervals, which correspond to the markers' levels. We want to accomplish a process of fuzzification of the measurable markers in order not to decide lengths of the level intervals intuitively (Rakus-Andersson, 2009, 2010a, 2010b, Rakus-Andersson & Jain, 2009).

A fuzzy set, say, *A* in the universe *X* is a collection of elements followed by the membership degrees that are computed by means of the membership function : [0,1] *<sup>A</sup> X* . Therefore *A* is denoted as {( , ( )), } *A x xxX <sup>A</sup>* . *A* is called normal if at least one element in the set *A* is assigned to the membership degree equal to 1. The support of *A* is a non-fuzzy set that consists of elements accompanied by membership degrees greater than 0.

The three quantitative markers *X*, *Y* and *Z* will be then differentiated into levels expressed by lists of terms. The terms from the lists are represented by fuzzy sets (Rakus-Andesson, 2007, 2010b), restricted by the membership functions lying over the domains *x x* min max , , *y y* min max , and *z z* min max , respectively.

In conformity with the physician's suggestions we introduce five levels of *X* , *Y* and *Z* as the collections

$$\mathbf{X} = \text{"age"} = \{ \mathbf{X}\_1 = \text{"very young"}, \text{"}\mathbf{X}\_2 = \text{"young"}, \mathbf{X}\_3 = \text{"middle-aged"}, \text{"}\mathbf{X}\_4 = \text{"old"}, \text{"}\mathbf{X}\_5 = \text{"very"} \}$$

$$\text{old"}\}$$

$$\text{Y = "CRP-value"} = \{\text{Y}\_1 = \text{"very low"}, \text{ Y }\_2 = \text{"low"}, \text{ Y }\_3 = \text{"medium"}, \text{ Y }\_4 = \text{"high"}, \text{ Y }\_5 = \text{"very"}\}$$

$$\text{high"})$$

and

530 New Advances in the Basic and Clinical Gastroenterology

inserting importance weights assigned to powerful biological indices taking place in the operation decision process. To compute the weights of importance the Saaty algorithm

We introduce the medical task to solve in Section 2. In order to establish the code systems for clinical data the fuzzification of biological markers is discussed in Section 3. In Section 4 we analyze the way of determining the patient characteristics, which should connect the mix of different codes in one value. The adaptation of the NS algorithm to surgery assumptions is made in Section 5. Finally, in Section 6 we test clinical data to prove the action of the

Gastric cancer patients are mostly cured by operating on them. Different types of surgery are taken into account. Two of them, namely, the partial resection surgery contra the radical surgery are considered by surgeons when evaluating biological markers in the context of their deviations from normal values (Do-Kyong Kim et al., 2009; de Mello et al., 1983).

Nevertheless, a surgeon often must decide if any operation on a patient is possible. The choice between the status "operate" and "do not operate" will constitute the main problem to solve by engaging different algorithms with their origins in Computational Intelligence. The selection will be made on the basis of three biological markers listed as *X* = *age*, *Y* = *CRP-value*  (C reactive proteins), and *Z* = *body weight* (Do-Kyong Kim et al., 2009; de Mello et al., 1983). These are considered as the most important indices in gastric cancer surgery decision making. As a leading method, which should provide us with decisions "operate" against "do not operate", we adapt the NS (Negative Selection) algorithm of immunological computation. To comprehend better some associations between the body immunological system and artificially invented algorithms based on the body protection system let us recall the most

Immunity refers to the condition, in which the organism can resist diseases. A broader definition of immunity is a reaction to foreign substances (pathogens). The biological immune system (BIS) has the ability to detect foreign substances and to respond them. One of the main capabilities of the immune system is to distinguish own body cells from foreign substances, which is called self/non-self discrimination (Dasgupta & Nino, 2008;

This particular ability is assigned to a special kind of lymphocytes called T-cells produced in the bone marrow. The T-cells can differentiate own body cells from pathogenic cells; therefore they play the role of detectors. Both own cells belonging to the self region and foreign pathogen cells forming non-self domain have their special characteristics given in

Let us adapt the meaning of distribution into self and non-self in the medical application sketched as follows. To make a decision concerning an individual patient we assign the immunological region of self to "operate", whereas the non-self field will be identified with

To be able to use the self/non-self discrimination, accomplished by the NS algorithm, we need to create vectors of coded patient data. The own fuzzy technique will be involved to

**2. The description of the medical objective in gastric cancer surgery** 

(Saaty, 1978) is adopted.

essential definitions of immunity.

Engelbrecht, 2007; Forrest et al., 1997).

the form of vectors of coded or measured properties.

"do not operate" (Rakus-Andersson, 2011).

model introduced in the paper as an applicable novelty.

$$\mathbf{Z} = \text{"body weight"} = \{ \mathbf{Z}\_1 = \text{"very underweightted"}, \text{ } \mathbf{Z}\_2 = \text{"underweigtedted"}, \text{ } \mathbf{Z}\_3 = \text{"normal"}, \text{ } \mathbf{Z}\_4 = \text{"over weight"}, \text{ } \mathbf{Z}\_5 = \text{"over weight"}, \text{ } \mathbf{Z}\_6 = \text{"over weight"}, \text{ } \mathbf{Z}\_7 = \text{"very vera weighted"} \}.$$

Selected Algorithms of Computational Intelligence in Gastric Cancer Decision Making 533

0 for ,

( 2) 2

*E m m*

( 1) 2 2

2 for ,

*w*

*w E m E m*

*w E m E m*

1 2 for ,

1 2 for ,

0 for .

lying on the membership level 1 will have the same lengths. Moreover, the last "*left*"

2 for ,

*w*

*Lm* should have the intersection point with "*in the middle*" function on the

*w E m E m*

2 2

*w*

*E m m*

( 2) ( 1) 2 2

*m m*

*w*

*w*

( 1)

(3)

*<sup>m</sup>* , will be thus equal to

1 2 *E m m*

*m* 2 *E m L m*

2

(4)

*Lm* is thus

of the

( 1)

. The

2 *Lm*

2

. To find a uniform slope of

 

*m*

( 1) ( 2) 2 2 ( 2) 2

*m m*

*Lm* we make suggestions that the top segments of functions

2

; therefore 1

after multiplying the length of the distinct upper

*Lm* . Due to the previously made assumption the functions 1

2 ( 1) *E m m m* 

*Lm* is planned to be placed in (min(*L*1), 1) then

( 1) 2( 1)

 

*E m m*

( 1) ( 1) 2( 1) 2

*m m*

*w*

*w*

( 1) ( 1)( 2) 2 2 ( 1) ( 1)( 2) 2 ( 1)

*m m m Em m m m*

 

<sup>2</sup> ,0.5 *E m m*

is evaluated to be ( 1)

( 1) ( 1)( 2)

. The membership function of 1

1 for ,

1 2 for ,

*w E m E m*

 

*w*

0 for .

2 for ,

*w*

*w E m Em m*

*Em Em m*

.

( 1) 2( 1) ( 1) ( 1)

 

*E m m E m m m*

2

2

( 1) ( 1)

*m m E m m m*

 

*w*

( 2) 2

*E m m E m E E m*

2

2

2

*<sup>L</sup> <sup>w</sup> <sup>E</sup> E m*

2

2

( 2) 2

*E m E m m E m*

1

For the "*leftmost*" family *L*1,..., <sup>1</sup>

*<sup>m</sup>* . Particularly, 1

"*in the middle*" function 1

difference between 1

*Lm* we determine 1 1

Since the beginning of 1

function 1

*E*

2 1 <sup>2</sup> <sup>1</sup> <sup>1</sup> *E m*

and 1 2

> 1 2

> > 2

expanded as

*w sw*

2

( )

*w*

2 2

*m E*

2

membership level 0.5. Each upper segment of *Lt*, *t* = 1,..., <sup>1</sup>

*Lm* should intersect each other in point ( 1)

2 *Lm* 

*L L mm mm*

2

( 1) ( 1) ( 1)( 2)

<sup>1</sup> ( 1)( 2 ) <sup>2</sup> 2 ( 1)

*<sup>m</sup> Em m*

 

*Em Em Em m*

*m m* 2 ( 1) 2 ( 1)

( 1) *m* 2( 1) *E m L m*

segment by the number of the last left function. We have already found ( 1)

2

 

2

and 1

2 *Lm* 

<sup>1</sup>

( )

*w*

2( 1) 2 2 ( 1) () 1 , , , *<sup>m</sup>*

*L m m mm*

*L*

2 2

   

To accomplish a formal mathematical design of level restrictions let us study the special own technique of their implementations (Rakus-Andersson, 2007, 2010b). In general, we suggest that the linguistic list of terms is converted to a sampling of fuzzy sets *L*1,…,*Lm*, where *m* is an odd positive integer. Each term is represented by the corresponding fuzzy set, whose restriction is supposed to be created as the common formula depending on the *l*th value, where *l* = 1,…,*m*. We assume that supports of restrictions ( ) *Ll w* , *l* = 1,…,*m*, will cover parts of the reference set *L* = [min(*L*1),max(*Lm*)], *w L*. We introduce *E* = *L* as the length of *L*.

We divide all expressions *Ll* in three groups, namely, a family of "*leftmost*" sets *L*1,…, <sup>1</sup> 2 *Lm* , the set 1 2 *Lm* "*in the middle*" and a collection of "*rightmost*" sets 3 2 *Lm* ,…,*Lm*. To design the membership functions of *Ll* the s-class function

$$s(w, \alpha, \beta, \gamma) = \begin{cases} 0 & \text{for} \quad w \le \alpha, \\ 2\left(\frac{w-\alpha}{\gamma-\alpha}\right)^2 & \text{for} \quad \alpha \le w \le \beta, \\ 1 - 2\left(\frac{w-\gamma}{\gamma-\alpha}\right)^2 & \text{for} \quad \beta \le w \le \gamma, \\ 1 & \text{for} \quad w \ge \gamma, \end{cases} \tag{1}$$

will be adopted. The point (*α*, 0) starts the graph of the s-function, whereas the point (, 1) terminates this graph. The parameter is found as the arithmetic mean of *α* and . In *w* = the s-function reaches the value of 0.5.

When designing parameters of each class function we want to consider the possibility to obtain the equal lengths of these parts of *Ll*'s supports, which assist membership values greater than or equal to 0.5. The parts are regarded as the important representatives of fuzzy sets as they possess the largest index of the relationship to the set. We thus determine the breadth of each *Ll* to be *<sup>E</sup> <sup>m</sup>* on the membership level equal to 0.5.

Let us first design the parameters of the membership function "*in the middle*". The function of 1 2 *Lm* is constructed as a -function

$$
\pi(w) = \begin{cases}
\text{s(}w, \alpha\_1, \beta\_1, \gamma\_1 = \alpha\_2) & \text{for} \quad w \le \gamma\_1 = \alpha\_2, \\
1 - \text{s(}w, \alpha\_2 = \gamma\_1, \beta\_2, \gamma\_2) & \text{for} \quad w \ge \gamma\_1 = \alpha\_2.
\end{cases} \tag{2}
$$

We suppose that 1 2 *Lm* will be a normal fuzzy set in 1 2 <sup>2</sup> *E* .

In order to guarantee the breadth *<sup>E</sup> <sup>m</sup>* on the membership level 0.5, function 1 2 *Lm* should take ( 1) 1 22 2 *E E E m m m* and ( 1) 2 22 2 *E E E m m m* . Since 1 2 *Lm* is expected to preserve the uniform and symmetric shape then ( 2) <sup>1</sup> 22 2 <sup>2</sup> *E E E m m m* and ( 2) <sup>2</sup> 22 2 <sup>2</sup> *E E E m m m* . We state 1 2 *Lm* 's formula as

To accomplish a formal mathematical design of level restrictions let us study the special own technique of their implementations (Rakus-Andersson, 2007, 2010b). In general, we suggest that the linguistic list of terms is converted to a sampling of fuzzy sets *L*1,…,*Lm*, where *m* is an odd positive integer. Each term is represented by the corresponding fuzzy set, whose restriction is supposed to be created as the common formula depending on the *l*th

cover parts of the reference set *L* = [min(*L*1),max(*Lm*)], *w L*. We introduce *E* = *L* as the

We divide all expressions *Ll* in three groups, namely, a family of "*leftmost*" sets *L*1,…, <sup>1</sup>

 

*w*

 

When designing parameters of each class function we want to consider the possibility to obtain the equal lengths of these parts of *Ll*'s supports, which assist membership values greater than or equal to 0.5. The parts are regarded as the important representatives of fuzzy sets as they possess the largest index of the relationship to the set. We thus determine the

*<sup>m</sup>* on the membership level equal to 0.5.

Let us first design the parameters of the membership function "*in the middle*". The function

( , , , ) for , ( ) 1 ( , , , ) for . *s w w*

 

*s w w*

1 11 2 1 2 2 122 1 2

> .

 

*w*

 

will be adopted. The point (*α*, 0) starts the graph of the s-function, whereas the point (

2

0 for ,

2 for ,

*w*

*w*

 *w*

*w*

is found as the arithmetic mean of *α* and

 

 

<sup>2</sup> 22 2 <sup>2</sup> *E E E m*

*E*

*<sup>m</sup>* on the membership level 0.5, function 1

2

and ( 2)

1 2 for , 1 for ,

 

2

2

*w* , *l* = 1,…,*m*, will

*Lm* ,…,*Lm*. To design the

(1)

. In *w* =

(2)

*Lm* should take

2 *Lm* 's

2

*Lm* is expected to preserve the uniform

. We state 1

*m m*

2 *Lm* ,

, 1)

value, where *l* = 1,…,*m*. We assume that supports of restrictions ( ) *Ll*

*Lm* "*in the middle*" and a collection of "*rightmost*" sets 3

length of *L*.

the set 1

2

membership functions of *Ll* the s-class function

terminates this graph. The parameter

the s-function reaches the value of 0.5.

breadth of each *Ll* to be *<sup>E</sup>*

We suppose that 1

1 22 2 *E E E m m m*

formula as

*Lm* is constructed as a

of 1 2

( , , ,)

*s w*

*w*

2

and ( 1)

and symmetric shape then ( 2)

In order to guarantee the breadth *<sup>E</sup>*

( 1)


2 22 2 *E E E m m m*

*Lm* will be a normal fuzzy set in 1 2 <sup>2</sup>

. Since 1

*m m*

<sup>1</sup> 22 2 <sup>2</sup> *E E E m*

$$\mu\_{\frac{l\_{m+1}}{2}}(w) = \begin{cases} 0 & \text{for} \quad w \le \frac{\mathbb{E}(m-2)}{2m}, \\ 2\left(\frac{w - \frac{\mathbb{E}(m-2)}{2m}}{\frac{\mathbb{E}}{m}}\right)^2 & \text{for} \quad \frac{\mathbb{E}(m-2)}{2m} \le w \le \frac{\mathbb{E}(m-1)}{2m}, \\ 1 - 2\left(\frac{w - \frac{\mathbb{E}}{2}}{\frac{\mathbb{E}}{m}}\right)^2 & \text{for} \quad \frac{\mathbb{E}(m-1)}{2m} \le w \le \frac{\mathbb{E}}{2}, \\ 1 - 2\left(\frac{w - \frac{\mathbb{E}}{2}}{\frac{\mathbb{E}}{m}}\right)^2 & \text{for} \quad \frac{\mathbb{E}}{2} \le w \le \frac{\mathbb{E}(m+1)}{2m}, \\ 2\left(\frac{w - \frac{\mathbb{E}(m+2)}{2m}}{\frac{\mathbb{E}}{m}}\right)^2 & \text{for} \quad \frac{\mathbb{E}(m+1)}{2m} \le w \le \frac{\mathbb{E}(m+2)}{2m}, \\ 0 & \text{for} \quad w \ge \frac{\mathbb{E}(m+2)}{2m}. \end{cases} \tag{3}$$

For the "*leftmost*" family *L*1,..., <sup>1</sup> 2 *Lm* we make suggestions that the top segments of functions lying on the membership level 1 will have the same lengths. Moreover, the last "*left*" function 1 2 *Lm* should have the intersection point with "*in the middle*" function on the membership level 0.5. Each upper segment of *Lt*, *t* = 1,..., <sup>1</sup> 2 *<sup>m</sup>* , will be thus equal to 2 1 <sup>2</sup> <sup>1</sup> <sup>1</sup> *E m E <sup>m</sup>* . Particularly, 1 2 ( 1) *m* 2( 1) *E m L m* after multiplying the length of the distinct upper segment by the number of the last left function. We have already found ( 1) 1 2 *E m m* of the "*in the middle*" function 1 2 *Lm* . Due to the previously made assumption the functions 1 2 *Lm* and 1 2 *Lm* should intersect each other in point ( 1) <sup>2</sup> ,0.5 *E m m* ; therefore 1 2 ( 1) *m* 2 *E m L m* . The difference between 1 2 *Lm* and 1 2 *Lm* is evaluated to be ( 1) 2 ( 1) *E m m m* . To find a uniform slope of 1 2 *Lm* we determine 1 1 2 2 ( 1) ( 1)( 2) *m m* 2 ( 1) 2 ( 1) *Em Em m L L mm mm* .

Since the beginning of 1 2 *Lm* is planned to be placed in (min(*L*1), 1) then <sup>1</sup> 2 ( 1) ( 1) ( 1)( 2) 2( 1) 2 2 ( 1) () 1 , , , *<sup>m</sup> Em Em Em m L m m mm w sw* . The membership function of 1 2 *Lm* is thus expanded as

$$\mu\_{L\_{\frac{m-1}{2}}}(w) = \begin{cases} 1 & \text{for} \quad w \le \frac{\mathbb{E}(m-1)}{2(m+1)},\\ 1 - 2\left(\frac{w - \frac{\mathbb{E}(m-1)}{2(m+1)}}{\frac{\mathbb{E}(m-1)}{m(m+1)}}\right)^2 & \text{for} \quad \frac{\mathbb{E}(m-1)}{2(m+1)} \le w \le \frac{\mathbb{E}(m-1)}{2m},\\ 2\left(\frac{w - \frac{\mathbb{E}(m-1)(m+2)}{2m(m+1)}}{\frac{\mathbb{E}(m-1)}{m(m+1)}}\right)^2 & \text{for} \quad \frac{\mathbb{E}(m-1)}{2m} \le w \le \frac{\mathbb{E}(m-1)(m+2)}{2m(m+1)},\\ 0 & \text{for} \quad w \ge \frac{\mathbb{E}(m-1)(m+2)}{2m(m+1)}. \end{cases} \tag{4}$$

Selected Algorithms of Computational Intelligence in Gastric Cancer Decision Making 535

(*t*), is authorized by the fact that *t* = 1 should be followed by

( 1)( 2) 2 ( 1)

 

*E t wE t*

) ( 1) 2 2( 1)

*m m*

( ) ( ),

*t wE t*

*E m*

 

 

*Em m m m*

( ) ( 1)( 2) ( 1) 2 ( 1) 2 ( )

( 1) 2( 1)

 

*E m m*

*wE t*

*w E t*

*t m m m*

2 for ( ) ( ),

The functions of fuzzy sets *L*1,...,*Lm* intend to maintain the same distances on the membership level 0.5. This property allows assigning to *L*1,...,*Lm* the relevant parts of their supports possessing the same length. The relevant parts of fuzzy sets consist of the sets' elements that reveal the membership degree values greater than or equal to 0.5. When forming the supports of the same length, in turn, we warrant the partition of [min(*L*1),max(*Lm*)] in equal subintervals standing for *Ll* levels, *l* = 1,...,*m*. Apart from that, the "*leftmost*" and "*rightmost*" functions also keep the same distances on the membership level 1.

All steps of the discussed algorithm, which initiates three sets of membership functions corresponding to a list of terms, can be sampled in the block scheme. We need to follow the steps of the scheme together with formulas (3), (5) and (7) to write the excerpt of a computer program. We emphasize that the only data, used in the algorithm, are the length of the reference set and the number of functions. We do not need to specify the sets' borders in the process of the program initialization, as most of programmers do, since the borders are computed automatically by formulas (3), (5) and (7). The steps of the algorithm flow chart

The procedure discussed above has started introduction of membership functions typical of

For five levels of *X* = [0, 100], *L* = *X*, *w* = *x*, *m* = 5, *E* = 100, the leftmost family is revealed by

1 for 33.33(0.5 ),

0 for 46.66(0.5 ),

1 2 for 33.33(0.5 ) 40(0.5 ),

*x t*

*x t*

*tx t*

(8)

*tx t*

2 for 40(0.5 ) 46.66(0.5 ),

*w E t Em m E m*

*<sup>m</sup>*<sup>1</sup> . Formula (7) constitutes a common base

 

(7)

comparing to the creation of

for deriving membership functions

<sup>3</sup> <sup>1</sup> <sup>2</sup>

*m t*

*L*

 

 

are sampled in Fig. 1.

for *t* = 1, 2 due to (5).

(1) = 1, whereas *t* = <sup>1</sup>

*<sup>m</sup>* is helped by

( <sup>1</sup> 2 *<sup>m</sup>* ) = <sup>2</sup>

0 for ( ),

2

( )

*w*

( 1)( 2) 2 ( 1) ( 1) ( 1)

 

*Em m m m E m m m*

2

2

*wE t E m*

1 for ( ).

This feature provides us with a harmonious arrangement of function shapes.

levels of *X*, *Y* and *Z* which, in turn, represent *age*, *CRP* and *body weight*.

*t*

*x t t*

<sup>2</sup> 33.33(0.5 ) 13.33(0.5 ) <sup>2</sup> 46.66(0.5 ) 13.33(0.5 )

*<sup>X</sup> x t*

( )

*x*

*t*

( ) ( 1

*E*

1 2 for

( )

*t*

( 1) 2( 1) ( 1) ( 1)

 

*E m m E m m m*

All constraints characteristic of the "*leftmost*" family of fuzzy sets will be given after inserting parameter <sup>2</sup> <sup>1</sup> ( ) *<sup>m</sup> t t* , *t* = 1,…, <sup>1</sup> 2 *<sup>m</sup>* , in (4) to form it as (Rakus-Andersson, 2007)

$$\mu\_{L\_t}(w) = \begin{cases} 1 & \text{for } \quad w \le \frac{E(m-1)}{2(m+1)}\delta(t), \\ 1 - 2\left(\frac{w - \frac{E(m-1)}{2(m+1)}\delta(t)}{\frac{E(m-1)}{m(m+1)}\delta(t)}\right)^2 & \text{for } \quad \frac{E(m-1)}{2(m+1)}\delta(t) \le w \le \frac{E(m-1)}{2m}\delta(t), \\ 2\left(\frac{w - \frac{E(m-1)(m+2)}{2m(m+1)}\delta(t)}{\frac{E(m-1)}{m(m+1)}\delta(t)}\right)^2 & \text{for } \quad \frac{E(m-1)}{2m}\delta(t) \le w \le \frac{E(m-1)(m+2)}{2m(m+1)}\delta(t), \\ 0 & \text{for } \quad w \ge \frac{E(m-1)(m+2)}{2m(m+1)}\delta(t). \end{cases} (5)$$

Parameter (*t*) takes the value of 1 for *t* = <sup>1</sup> 2 *<sup>m</sup>* , which means that ( <sup>1</sup> 2 *<sup>m</sup>* ) in (5) has no influence on the shape of the last left function. However, the introduction of (*t*) in (5) induces the narrowing effects in the supports of the other left function shapes. To preserve the same lengths of upper segments corresponding to membership 1 and middle segments attached to membership 0.5 we adjust (*t*), assisting the left function *Lt*, to be equal to 1 2 1 *m*

multiplied by the function number *t*.

In order to start the implementation of the "*rightmost*" family functions let us note that the first right function 3 2 ( ) *Lm w* should possess 1 2 ( ) *Lm w* 's inverted shape. We generate the membership function of 3 2 *Lm* by

$$
\mu\_{L\_{\frac{m+1}{2}}}(w) = \begin{cases}
0 & \text{for} \quad w \le E - \frac{\mathbb{E}(m-1)(m+2)}{2m(m+1)}, \\
2\left(\frac{w - \left(E - \frac{\mathbb{E}(m-1)(m+2)}{2m(m+1)}\right)}{\frac{\mathbb{E}(m-1)}{m(m+1)}}\right)^2 & \text{for} \quad E - \frac{\mathbb{E}(m-1)(m+2)}{2m(m+1)} \le w \le E - \frac{\mathbb{E}(m-1)}{2m}, \\
1 - 2\left(\frac{w - \left(E - \frac{\mathbb{E}(m-1)}{2(m+1)}\right)}{\frac{\mathbb{E}(m-1)}{m(m+1)}}\right)^2 & \text{for} \quad E - \frac{\mathbb{E}(m-1)}{2m} \le w \le E - \frac{\mathbb{E}(m-1)}{2(m+1)}, \\
1 & \text{for} \quad w \ge E - \frac{\mathbb{E}(m-1)}{2(m+1)}.
\end{cases} (6)$$

The function of 3 2 *Lm* is symmetrically inverted to the function of 1 2 *Lm* over interval [min(*L*1), max(*Lm*)].

Hence, the membership function 1 2 ( 1) ( 1) ( 1)( 2) 2( 1) 2 2 ( 1) () 1 , , , *<sup>m</sup> Em Em Em m L m m mm w sw* will be changed into 3 2 ( 1)( 2) ( 1) ( 1) 2 ( 1) 2 2( 1) () , , , *<sup>m</sup> Em m Em Em L m m m m w s wE E E* .

To generate the "*rightmost*" family of sets 3 2 *Lm* ,...,*Lm* we need to create a new parameter 2 <sup>1</sup> ( ) 1 ( 1) *<sup>m</sup> t t* , *t* = 1,..., <sup>1</sup> 2 *<sup>m</sup>* , which will be inserted in (6). The construction of (*t*), when comparing to the creation of (*t*), is authorized by the fact that *t* = 1 should be followed by (1) = 1, whereas *t* = <sup>1</sup> 2 *<sup>m</sup>* is helped by ( <sup>1</sup> 2 *<sup>m</sup>* ) = <sup>2</sup> *<sup>m</sup>*<sup>1</sup> . Formula (7) constitutes a common base for deriving membership functions

$$\begin{cases} \begin{aligned} &w\_{\frac{m+1}{2},\epsilon-1}(w) = \\ &\begin{cases} 0 & \text{for} \quad w \leq E - \frac{E(m-1)(m+2)}{2m(m+1)}\varepsilon \ell(t), \end{cases} \\ &2\left(\frac{w-\left(E-\frac{E(m-1)(m+2)}{2m(m+1)}\varepsilon\ell(t)\right)}{\frac{E(m-1)}{m(m+1)}\varepsilon\ell(t)}\right)^{2} &\text{for} \quad E - \frac{E(m-1)(m+2)}{2m(m+1)}\varepsilon\ell(t) \leq w \leq E - \frac{E(m-1)}{2m}\varepsilon\ell(t), \\ &1 - 2\left(\frac{w-\left(E-\frac{E(m-1)}{2(m+1)}\varepsilon\ell(t)\right)}{\frac{E(m-1)}{m(m+1)}\varepsilon\ell(t)}\right)^{2} &\text{for} \quad E - \frac{E(m-1)}{2m}\varepsilon\ell(t) \leq w \leq E - \frac{E(m-1)}{2(m+1)}\varepsilon\ell(t), \\ &1 &\text{for} \quad w \geq E - \frac{E(m-1)}{2(m+1)}\varepsilon\ell(t). \end{cases} \end{cases} \tag{7}$$

The functions of fuzzy sets *L*1,...,*Lm* intend to maintain the same distances on the membership level 0.5. This property allows assigning to *L*1,...,*Lm* the relevant parts of their supports possessing the same length. The relevant parts of fuzzy sets consist of the sets' elements that reveal the membership degree values greater than or equal to 0.5. When forming the supports of the same length, in turn, we warrant the partition of [min(*L*1),max(*Lm*)] in equal subintervals standing for *Ll* levels, *l* = 1,...,*m*. Apart from that, the "*leftmost*" and "*rightmost*" functions also keep the same distances on the membership level 1. This feature provides us with a harmonious arrangement of function shapes.

All steps of the discussed algorithm, which initiates three sets of membership functions corresponding to a list of terms, can be sampled in the block scheme. We need to follow the steps of the scheme together with formulas (3), (5) and (7) to write the excerpt of a computer program. We emphasize that the only data, used in the algorithm, are the length of the reference set and the number of functions. We do not need to specify the sets' borders in the process of the program initialization, as most of programmers do, since the borders are computed automatically by formulas (3), (5) and (7). The steps of the algorithm flow chart are sampled in Fig. 1.

The procedure discussed above has started introduction of membership functions typical of levels of *X*, *Y* and *Z* which, in turn, represent *age*, *CRP* and *body weight*.

For five levels of *X* = [0, 100], *L* = *X*, *w* = *x*, *m* = 5, *E* = 100, the leftmost family is revealed by

$$\mu\_{X\_i}(\mathbf{x}) = \begin{cases} 1 & \text{for } \quad \mathbf{x} \le 33.33(0.5t), \\ 1 - 2\left(\frac{\mathbf{x} - 33.33(0.5t)}{13.33(0.5t)}\right)^2 & \text{for } \quad 33.33(0.5t) \le \mathbf{x} \le 40(0.5t), \\ 2\left(\frac{\mathbf{x} - 46.66(0.5t)}{13.33(0.5t)}\right)^2 & \text{for } \quad 40(0.5t) \le \mathbf{x} \le 46.66(0.5t), \\ 0 & \text{for } \quad \mathbf{x} \ge 46.66(0.5t), \end{cases} \tag{8}$$

for *t* = 1, 2 due to (5).

534 New Advances in the Basic and Clinical Gastroenterology

All constraints characteristic of the "*leftmost*" family of fuzzy sets will be given after

( 1) 2( 1)

 

*E m m*

*w t*

( ) ( 1) ( 1)( 2) 2 2 ( 1) ( )

> ( 1)( 2) 2 ( 1) or ( ). *Em m m m w t*

*<sup>m</sup>* , which means that

( ) ( 1) ( 1) 2( 1) 2 ( )

*t m m*

1 2 for ( ) ( ),

 

*w t E m Em m t m m m*

2

induces the narrowing effects in the supports of the other left function shapes. To preserve the same lengths of upper segments corresponding to membership 1 and middle segments

In order to start the implementation of the "*rightmost*" family functions let us note that the

0 for ,

2 ( ) *Lm* 

2 for ,

*w E*

*w E Em m E m*

1 2 for ,

*w sw*

( 1)( 2) ( 1) ( 1)

2

 .

*Em m Em Em*

*w E E m E m*

*w E*

*w t E m E m*

2 for ( ) ( ),

*<sup>m</sup>* , in (4) to form it as (Rakus-Andersson, 2007)

*tw t*

 

*t w t*

(*t*), assisting the left function *Lt*, to be equal to 1

( 1)( 2) 2 ( 1)

 

*Em m m m*

*E w E*

( 1)( 2) ( 1) 2 ( 1) 2

*m m m*

2

*Em Em Em m*

( 1) ( 1) ( 1)( 2)

*Lm* ,...,*Lm* we need to create a new parameter

 will be

*Lm* over interval [min(*L*1),

(*t*), when

( 1) ( 1) 2 2( 1)

*m m*

(

 1) 2( 1) . *<sup>m</sup>* 

*E wE*

*E m*

2( 1) 2 2 ( 1) () 1 , , , *<sup>m</sup>*

*L m m mm*

*<sup>m</sup>* , which will be inserted in (6). The construction of

 

( <sup>1</sup> 2

*w* 's inverted shape. We generate the

  (5)

*<sup>m</sup>* ) in (5) has no

(*t*) in (5)

(6)

2 1 *m*

2

1 for ( ),

2

2

influence on the shape of the last left function. However, the introduction of

*w* should possess 1

changed into 3

2

*w s wE E E*

*Em m m m E m m m*

( 1)( 2 ) 2 ( 1) ( 1) ( 1)

 

2

2

*Lm* is symmetrically inverted to the function of 1

Hence, the membership function 1 2

2 ( 1) 2 2( 1) () , , , *<sup>m</sup>*

*L m m m m*

*m E m m m*

 

( 1) ( 1)

1 for

inserting parameter <sup>2</sup>

( )

*w*

attached to membership 0.5 we adjust

( )

2

2

To generate the "*rightmost*" family of sets 3

*t t* , *t* = 1,..., <sup>1</sup>

2 <sup>1</sup> ( ) 1 ( 1) *<sup>m</sup>* *w*

2 ( ) *Lm* 

> 

2 *Lm* by

<sup>3</sup> ( 1) <sup>2</sup> 2( 1)

*<sup>m</sup> E m*

multiplied by the function number *t*.

*L*

first right function 3

membership function of 3

*L*

The function of 3

max(*Lm*)].

Parameter

<sup>1</sup> ( ) *<sup>m</sup>* 

> 

*<sup>t</sup> Em m*

*t t* , *t* = 1,…, <sup>1</sup>

( 1) 2( 1) ( 1) ( 1) ( 1)( 2 ) 2 ( 1) ( 1) ( 1)

 

*E m m E m m m*

*m m E m m m*

(*t*) takes the value of 1 for *t* = <sup>1</sup>

 

0 f

Selected Algorithms of Computational Intelligence in Gastric Cancer Decision Making 537

0 for 30,

 

2 for 30 40,

*x*

*x*

*x*

*x*

(10)

x

*x*

*x*

1 2 for 40 50,

1 2 for 50 60,

2 for 60 70, 0 for 70.

The "*in the middle*" *X*-level "*middle-aged*" has, in accord with (3), the constraint

*x*

<sup>2</sup> <sup>30</sup> 20

*x*

*x*

3

membership degrees greater than or equal to 0.5.

part of *X*3. The insertion of *t* = 2 in (9) produces a joint of

All levels of *X* are sketched in Fig. 2.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fig. 2. The fuzzy sets *X*1–*X*5

µ(x)

( )

*x*

*<sup>X</sup> <sup>x</sup>*

The parts of *X*1–*X*5 supports should be consisted of elements, which have the strongest connections with the *X*1–*X*5 fuzzy sets. Therefore we only select the elements having the

To make the partition of *X* in subintervals representing levels *X*1–*X*5 we return to formulas (8), (9) and (10). Due to (8), to find the subinterval of *X* assisting *X*1 when *t* = 1, we concatenate the intervals *x* 33.33(0.5 1) and 33.33(0.5 1) 40(0.5 1), *x* leading to [0, 20]. We have chosen these intervals, which contain elements of *X*1 furnished with membership degrees greater than or equal to 0.5. For *t* = 2, set in two first intervals of (8), we aggregate *x* 33.33(0.5 2) and 33.33(0.5 2) 40(0.5 2) *x* in [0, 40]. This generates the interval [0, 40]–[0, 20] = [20, 40] typical of *X*2. By (10) we find [40, 60] = [40, 50] + [50, 60] as an essential

"*very young*" "*young*" "*middle aged*" " *old* " "*very old*"

0 10 20 30 40 50 60 70 80 90 100

Fig. 1. The flow chart of the *L*1,…,*Lm* implementation

The rightmost family of *X*-levels, composed with conformity with (7), is stated as

$$\begin{cases} \mu\_{X\_{4 \leftrightarrow 1}}(\mathbf{x}) = \\ \begin{cases} 0 \text{ for } \mathbf{x} \le 100 - 46.66(1 - 0.5(t - 1)), \\ 2\left(\frac{x - (100 - 46.66(1 - 0.5(t - 1)))}{13.33(1 - 0.5(t - 1))}\right)^2 \\ \text{for } 100 - 46.66(1 - 0.5(t - 1)) \le x \le 100 - 40(1 - 0.5(t - 1)), \\ 1 - 2\left(\frac{x - (100 - 33.33(1 - 0.5(t - 1)))}{13.33(1 - 0.5(t - 1))}\right)^2 \\ \text{for } 100 - 40(1 - 0.5(t - 1)) \le x \le 100 - 33.33(1 - 0.5(t - 1)), \\ 1 \text{ for } x \ge 100 - 33.33(1 - 0.5(t - 1)), \end{cases} \tag{9}$$

for *t* = 1, 2.

The "*in the middle*" *X*-level "*middle-aged*" has, in accord with (3), the constraint

$$\mu\_{X\_3}(\mathbf{x}) = \begin{cases} 0 & \text{for} \quad \mathbf{x} \le 30, \\ 2\left(\frac{\mathbf{x} - 30}{20}\right)^2 & \text{for} \quad 30 \le \mathbf{x} \le 40, \\ 1 - 2\left(\frac{\mathbf{x} - 50}{20}\right)^2 & \text{for} \quad 40 \le \mathbf{x} \le 50, \\ 1 - 2\left(\frac{\mathbf{x} - 50}{20}\right)^2 & \text{for} \quad 50 \le \mathbf{x} \le 60, \\ 2\left(\frac{\mathbf{x} - 70}{20}\right)^2 & \text{for} \quad 60 \le \mathbf{x} \le 70, \\ 0 & \text{for} \quad \mathbf{x} \ge 70. \end{cases} \tag{10}$$

All levels of *X* are sketched in Fig. 2.

536 New Advances in the Basic and Clinical Gastroenterology

YES

The rightmost family of *X*-levels, composed with conformity with (7), is stated as

for 100 46.66(1 0.5( 1)) 100 40(1 0.5( 1)),

3.33(1 0.5( 1)),

*tx t*

NO

*t*

(9)

NO

YES

Fig. 1. The flow chart of the *L*1,…,*Lm* implementation

( )

*x*

4 1

*X <sup>t</sup>*

 

 

2

1 2

for *t* = 1, 2.

*x t*

 

<sup>2</sup> 100 46.66(1 0.5( 1)) 13.33(1 0.5( 1))

*t*

0 for 100 46.66(1 0.5( 1)),

*x t*

*x t*

 

<sup>2</sup> 100 33.33(1 0.5( 1)) 13.33(1 0.5( 1))

1 for 100 33.33(1 0.5( 1)),

*x t*

*t*

for 100 40(1 0.5( 1)) 100 3

*t x*

The parts of *X*1–*X*5 supports should be consisted of elements, which have the strongest connections with the *X*1–*X*5 fuzzy sets. Therefore we only select the elements having the membership degrees greater than or equal to 0.5.

To make the partition of *X* in subintervals representing levels *X*1–*X*5 we return to formulas (8), (9) and (10). Due to (8), to find the subinterval of *X* assisting *X*1 when *t* = 1, we concatenate the intervals *x* 33.33(0.5 1) and 33.33(0.5 1) 40(0.5 1), *x* leading to [0, 20]. We have chosen these intervals, which contain elements of *X*1 furnished with membership degrees greater than or equal to 0.5. For *t* = 2, set in two first intervals of (8), we aggregate *x* 33.33(0.5 2) and 33.33(0.5 2) 40(0.5 2) *x* in [0, 40]. This generates the interval [0, 40]–[0, 20] = [20, 40] typical of *X*2. By (10) we find [40, 60] = [40, 50] + [50, 60] as an essential part of *X*3. The insertion of *t* = 2 in (9) produces a joint of

Fig. 2. The fuzzy sets *X*1–*X*5

Selected Algorithms of Computational Intelligence in Gastric Cancer Decision Making 539

In order to measure the affinity (coverage) of two code vectors *v*1 and *v*2 of the same length over the same alphabet we are furnished with the *r*-contiguous bit matching rule, which

provides us with a true match(*v*1, *v*2) if *v*1 and *v*2 agree in *r* contiguous locations.

**4. The selection of the most representative data vectors for the decision** 

We have already mentioned that we need the "operate" types of patient data vectors as the entries of the NS discrimination algorithm. We thus want to prepare typical data strings for

Let us first treat the vector *v* = (*x*, *y*, *z*) as the string of integers *v* = (*x y z*), where *x*, *y* and *z* can take the code values 0, 1, 2, 3, 4. We form the function *f*(*x y z*) = *x* + *y* + *z* to measure the common code value of the data vector. To make the selection of "operate" type vectors even more accurate let us assign the weights of power-importance to the biological indices considered in the operation decision. In the gastric cancer operation decision we first concentrate our attention on the changes of *CRP*- values, which points out *CRP* as the most decisive factor. The analysis of *CRP* is followed by the judgment of age and, finally, we check the values of body weights. Hence, we state the ranking of the symptom importance

A procedure for obtaining a ratio scale of importance for a group of *m* elements (in the considered case – biological markers) was developed by Saaty (Saaty, 1978). Assume that we have *m* objects (symptoms) and we want to construct a scale, rating these objects as to their importance with respect to the decision. We ask a decision-maker to compare the objects in paired comparison. If we compare object *j* with object *k*, *j*, *k* = 1,...,*m*, then we will assign the

2. If objective *j* is more important than objective *k* then *bjk* gets assigned a number

Having obtained the above judgments an *<sup>m</sup> <sup>m</sup>*importance matrix , 1

constructed. The importance weights are decided as components of this eigenvector that

*m jk <sup>j</sup> <sup>k</sup> B b* is

as *CRP age body weight* , provided that means "more important than".

For *v*1 = (3, 1, 3) and *v*2 = (3, 1, 2), when *r* = 2, match(*v*1, *v*2) is true.

**Example 2** 

**"operate"** 

the decision "operate" in advance.

values *bjk* and *bkj* as follows

according to the following scheme:

*expressed by the value of bjk*

*Intensity of importance Definition* 

 1 Equal importance of *xj* and *xk* 3 Weak importance of *xj* over *xk* 5 Strong importance of *xj* over *xk* 7 Demonstrated importance of *xj* over *xk* 9 Absolute importance of *xj* over *xk*

If object *k* is more important than object *j*, we assign the value of *bkj*.

corresponds to the largest in magnitude eigenvalue of the matrix *B*.

2, 4, 6, 8 Intermediate values

*jk*

*<sup>b</sup>* .

1. <sup>1</sup> *kj*

*b*

100 40(1 0.5(2 1)) 100 33.33(1 0.5(2 1)) *x* and *x* 100 33.33(1 0.5(2 1)) to be a common interval [80, 100] regarded as the domain of *X*5. By setting *t* = 1 in the last intervals of (9) we get the field [60, 100]. It means that *X*4 will be given by [60, 80] = [60, 100]–[80, 100]. We are furnished with the same intervals after accomplishing the close analysis of Fig. 2 on the membership level 0.5.

Let us now initiate the associations among the terms of *X*, characteristic intervals of these terms and assigned to them codes due to the scheme


We emphasize the role of an elegant mathematical design of *X*'s membership functions, which allows making the partition of the *X*-domain in equal intervals. Definitely, we obtain the same results when dividing the length of *X* by the number of levels to get a length of one part but the effects computed by means of membership functions only confirm this intuitive calculation. Moreover we can modify the arbitrary lengths of *X*-subintervals by making changes in the formulas of (*t*) and (*t*).

By applying the same technique to *Y* = [0, 60], *L* = *Y*, *w* = *y*, *m* = 5, *E* = 60 we generate the code pattern


Lastly, if *Z*= [40, 120] for men, *L* = *Z*, *w* = *z*, *m* = 5, *E* = 80 then


If we collect clinical data, concerning a patient examined then we will be now capable to create code vectors taking place in the discrimination NS algorithm.

#### **Example 1**

An eighty one-year-old man, whose *CRP* is 17 and weight is 91, will be given by the vector *v* = (4, 1, 3).

In order to measure the affinity (coverage) of two code vectors *v*1 and *v*2 of the same length over the same alphabet we are furnished with the *r*-contiguous bit matching rule, which provides us with a true match(*v*1, *v*2) if *v*1 and *v*2 agree in *r* contiguous locations.
