THE USE OF PARTIAL COEFFICIENTS TO CONSIDER

THE INCIDENCES OF HIDDEN UNCERTAINTIES

Arturo J Bignoli

Academia Nacional de Ingenieria

(Argentina)

  1. Introduction

Codes of practice qualify in linguistic form hidden uncertainties and assign them partial numerical coefficients according to their magnitude.

These coefficients enlarge or diminish the values of variables given by the codes, according to the accomplishment of the preconditions established by them.

The linguistic qualification of uncertainties is given by adjectives, modified or not by adverbs; and therefore the partial coefficients are numerical expressions of those adjectives and adverbs.

Fuzzy Logic and Fuzzy Arithmetic will be used to obtain the values of those partial coefficients.

  1. The scales of fuzzy numerical values of linguistic qualifications

We use “fuzzy triangular numbers” (F.T.N.) See fig. 1

x

      

F.N=  (xi) / xi

i

F.N. = 0 |(n-2) + 0.5| (n – )| + 1|(n) + 0,5| (n+) + 0 |(n+2)

Fig. 1

x* is the preferred value with support x = 1, also called mode

xs and xi are the superior and inferior values accepted for x, with support zero.

xs – xi = r (x*) is the range that includes x* and represents the uncertainty of the value xi*

We adopt for qualifications one of the two following scales. The first one is called “strong” scale, and the second, “soft” scale. The “similitude” between 2 successive values in the strong scale is smaller than the one in the soft scale, as it may be observed. In the first scale the rank is 2 and in the second 4. We adopt  = 1 for this example. The basic variable x may have eleven values from 0 to 10.

Strong Scale for  = 1Soft Scale for  = 1

X* / xi / x* / xs / Literary Value / x* / xi / x* / xs / Literary Value
10 / 9 10 11 / Big + / 10 / 8 10 12 / Big +
9 / 8 9 10 / Big / 9 / 7 9 11 / Big
8 / 7 8 9 / Big - / 8 / 6 8 10 / Big -
7 / 6 7 8 / Big/Med / 7 / 5 7 9 / Big/Med
6 / 5 6 7 / Med. + / 6 / 4 6 8 / Med. +
5 / 4 5 6 / Medium / 5 / 3 5 7 / Medium
4 / 3 4 5 / Med. - / 4 / 2 4 6 / Med. -
3 / 2 3 4 / Med./Small / 3 / 1 3 5 / Med./Small
2 / 1 2 3 / Small + / 2 / 0 2 4 / Small +
1 / 0 1 2 / Small / 1 / -1 1 3 / Small
0 / -1 0 1 / Small - / 0 / -2 0 2 / Small -

 x*; r= 2 for =1 x*; r= 4 for =1

In general:In general:

xi = x* - ; xs = x* + xi = x* - 2; xs = x* + 2

r (x) = xs – xi = 2 r (x) = xs – xi = 4 

The similitude between two following values for =1 are:

a)In Strong Scale: (fig. 2)

 (x)

1|(n-)  (n)| = 0.5/2 = 0.25

|(n-)| = |n| = 2 /2 = 

|(n-)  n| =  +  – 0.25 = 1.75

0.5

   

0

n-2 n- n n+ x

sim [(n-); n] = 0.25 /1.75 = 0.143

b) In Soft Scale (fig. 3)

1.00

|(n – )  n| = 0.75x 3/2 = 1.125

0.75|(n-)| = |n| = 4/2 = 2

| (n-)  n| = 2+2- 1.125 = 2.875

0.50

0.25

0      x

(n-3) (n-2) (n-) n (n+) (n+2)

sim [(n-); (n)] = 1.125/2.875 = 0.391

c) In strong scale two following qualifications are 0.391/0.143 = 2.73 times more dissimilar than in softscale. So, the strong scale is more sensible than the soft one.

3. The calculation of values of partial coefficients by “filtering”

“Filters” are the particular fuzzy numbers with the shapes shown in fig. 4 a and b.

(x)(x)

1F F 1 F F

0.75 0.75

0.5 0.50

0.25 0.25

          

0 x* xs x 0 xi x*x

Not bigger than x*Not smaller than x*

If x x* (x) = 1If x x* (x) = 1

If x* x xs (x) = 1/2  (xs-x)If xi x x* (x)= (x-xi)/2

If x x* (x)= 0If x xi(x)=0

4.a4.b

Fig. 4

When qualifying magnitudes with fuzzy numbers and considering that they may be “pro-safety” for instance, resistance R, or “anti-safety”, for instance forces S, we can use for “pro-safety” magnitudes R filters “not smaller than” (fig. 4b) and for “anti-safety” magnitudes S filters “not bigger than” (fig. 4a)

Then, there will be two possible cases:

a)magnitudes “anti-safety” (fig. 5)

(x)

1

0.75

0.5

0.25

      

0

xis x*s xs x

xiF xF*

(n-2) (n-) n (n+) (n+2)

|S(n-2) F(S)| = 2 (1/2)2 = 0.5

|S(n-2) F(S)| / |S(n-2)| = 0.5/2 = 0.25 (not accept part of S(n-2)

We adopt s = 1 + (|S F|/|S|), so in the case of “soft scale”( figure 5):

s = 1 + 0.25 = 1.25

For S(n-) we find s =1 + 0.56 = 1.56

S(n) s= 1+1 = 2

S(n-3) s= 1+ 0.06 = 1.06

S(n - 4) s= 1 + 0 = 1

With the same filter, we obtain for hard scale

(n): s =2

(n-): s =1.333

(n-2): s = 1.166

(n-3):s = 1

2

For “soft” scale

1.75

For “hard” scale

1.5

1.25

1     x

(n-4) (n-3) (n-2) (n-) (n) (n+)

Fig. 6

b)Magnitudes “pro-safety”

We adopt R = 1 - (|R F|/|R|)

Observing fig. 7 it is easy to deduce that for “soft” scale it is:

(n):R = 0

(n+):R = 0.44

(n+2):R = 0.75

(n+3):R = 0.94

(n+4):R = 1

And for “hard” scale:

(n):R = 0

(n+):R = 0.666

(n+2):R = 0.833

(n+3):R = 1

(x)

1

0.75

0.5

0.25

0

Fig. 7

All the information is gathered as in Fig. 8:

s; R

2

1.75

1.50

1.25

1R

0.75

0.50

0.25

0

        x*

| |

sensitivity zone + 4 

Fig. 8

4. The practical sequence for obtaining R or s for a literary variable X

1st Experts after analyzing the concept of every variable assign by consensus a F.N. to express his literary qualification

2ndAlso by consensus, they choose a value of for every variable

3rdAgain by consensus, they choose the filter convenient for every variable or group of variables.

4th It is very easy then (a simple program can do it) to calculate s and R.

It should remain clear that choosing the value of  we also choose the range of the filter (2 or 4). So, the bigger  means more uncertainty in the fuzzy numbers but also a softer filter to obtain s and R. This is not convenient. The following figs. 10 (a) and (b) show how schanges with r(n) even with the same filter

s s

2 NF 2 N F

1.5 1.5

1 1

soft scale for  soft scale

|NF|/|F|=0.10filter |NF|/|F|=0.167for filter

s = 1.10 s = 1.167

r (F) = 2r(N) =  = 1r(F) = 2 r(N)=2=2

(N) = [(n-1/2); (n); (n+1/2)] (N)= [(n-1); (n); (n+1)]

Fig. 10 (a)Fig. 10(b)

With (N) = [(n-2); (n); (n+2)], = 1 and r(N) = 4, we have:

s n-2 n- n F

2

1.75 = 1

1.50

1.25r(F) = 2

1

0 0.5 1 1.5 2 sr(N)

Fig. 11Fig. 12

With a “softer filter”, the values of s are smaller:

s

2 F

1.75 = 1

1.50

1.25r(F) = 3

1

0 0.5 1 1.5 2 sr(N)

fig. 13fig. 14

6. Maximum and minimum values of S and R

As we saw, the values of coefficients correspond to two formulas:

Coefficient to decrease magnitudes “pro-safety” R

Coefficient to increase magnitudes “anti-safety” s

(x) (x)

1 SFs1 FR R

0x0x

“Not more than”“Not less than”

sR

Fig. 15 (a)Fig. 15 (b)

s = 1 + [|SFs| / |S|] 1s 2

R = 1 - [|RFr| / |R|] 0R 1

Maximum Values of S

(x)

(2)

max S

(1)

r(F) x*s x

Fig. 16

Any (FN) with sri(S) = r(F) gives s= 2 because SFs and then |FsS| / |S|=1 and s= 2

Values sri (S) > r(F) give s 2 (cases 1 or 2 in fig 16)

Minimum Values of R

(x)

(2)

minR (3) (4)

(1) r(F)

x*(Fr) x

Fig. 17

Any (FN) with x*(Fr) = x*(R) and srs(R) = r(Fr) gives R=1. Values with r(F) <srs(R) (cases 3 or 4 in fig 17) give values of R decreasing with increments of srs(R) because |RFr| does not change and |R| is increased.

7. Conclusions

We saw that it is possible to obtain values of partial coefficients in order to take into account the hidden uncertainties, literally qualified.

The procedure proposed seems conveniently rational and easy to manage.

Experts or code makers are required to transform subjective opinions that result of uncertainties in a consensus that is objective, then partial coefficients are calculated.

Anyhow the dependence of subjectivity exists, but having much smaller incidence. The independence of subjectivity seems impossible to reach. It is possible only in a deterministic problem with prestablished hypotheses.

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