Risk Evaluation Model
based on triangular fuzzy
membership function
Abstracting from Mai Haiming (2009). Model of
evaluation for occupational disease hazard risks
based on fuzzy membership functions. Chinese
Journal of Industrial Hygiene and Occupational
Disease，27(5):298-302.
Modeling
 Let domain D belong to range zero to one
(D=[0,1]) and function P:D→D is the hazard
degree of a sort of poison (chemical) and P
is monotonic increasing function.
 Let membership of the total risk U=(u1,u2,u3),
ui(R):D→D，i=1,2,3. u1 belong to high risk
and u2 belong to middle risk and u3 belong
to low risk.
Modeling
u1
1.0
u2
u3
0.5
0.0
0.5
1.0
Figure 1. triangular fuzzy membership function of risk
R
Modeling
 Let p1 belong to the high hazard chemical
and p2 belong to the middle hazard
chemical and p3 belong to the low hazard
chemical.
 Let ph∈D is pmin∈p1 and pl∈D is pmax∈p3.
Modeling
 Let function of using factor of poison
C(c1,c2,…,cn)＝w1c1+ w2c2+…+ wncn：
Dn→D, ci, i＝1,2…n，wi≥0，i＝1，2…n is
weight, and ∑wi＝1.
 Let R(P, C):D2→D is risk and R about P is
monotonic increasing function and R about
C is monotonic decreasing function.
Modeling
 Since u1 is monotonic increasing function
 Hence If R=R(ph,1) then u1=0; if R>R(ph,1)
then u1>0; if R=1 then u1=1;
 Since u3 is monotonic decreasing function
 Hence if R=0 then u3=1; if R>0 then u3<1; if
R=R(pl,0) then u3=0；
Modeling
u1
1.0
u2
u3
0.5
0.0
0.5
1.0
Figure 1. triangular fuzzy membership function of risk
R
Modeling
 Let rm is the vertical projection point of
intersection between u1 and u3 on R(rm∈D);
if R=rm then u2=1 and if R<rm then u2 is
monotonic increasing function and if R>rm
then u2 is monotonic decreasing function.
Modeling
u1
1.0
u2
u3
0.5
0.0
0.5
1.0
Figure 1. triangular fuzzy membership function of risk
R
Modeling
 Let

R=wP + (1 - w)(1 - C)
(1)
 In this formula , the w is weight of P and
(1 – w) is weight of C; then

R(ph,1) =wph
(2)

R(pl,0)=1 + w(pl - 1)
(3)
Modeling

(4)
Modeling
u1
1.0
u2
u3
0.5
0.0
0.5
1.0
Figure 1. triangular fuzzy membership function of risk
R
Modeling
 Determine the function of U by using
triangular fuzzy membership.
 The general expression of triangular fuzzy
membership function is

(5)
Modeling
 The field of definitions is determined by
vector x(x=R) ; shape of curves is
determined by parameters a,b,c which are 3
vertexes of a triangle respectively and
a≤b≤c.
Modeling
 Based on formula (5): The expressions of
triangular fuzzy membership function about
U is:

(6)

(7)

(8)
Modeling
 Based on formula (2): let u1(x)=0 then

a1= R(ph,1) =wph
(9)
 Based on formula (3): let u3(x)=0 then

c3= R (pl,0)=1 + w(pl - 1)
(10)
 If intersection between u1 and u3 then
u1(rm)=u3(rm), u2=1, b2=rm and then

(11)
Modeling
 or x=c3／(1 - a1 + c3) and then

b2=c3／(1 - a1 + c3)
(12)
 put a1 and c3 into formula(12) and then

(13)
 Based on formula(4):
 if x1=wph + (1 - 0.5)(1 - w) then there is a
expression
(14)
Modeling
 if x2=wpl + (1 - 0.5)(1 - w) then there is a
expression

(15)
 ph∈[0,1] and pl∈[0,1]. Determine the model
function by using various range of ph and pl.
set Err is error:

(16)
Modeling
 After value of ph and pl have been determined,
if put ph and pl into formula(12) and formula (13)
and formula (10) then a1 and b2 and c3 can be
found out and then if put a1 and b2 and c3 into
formula(16) and then given the initial value of a2
and c2 and w respectively randomly and then
value of a2 and c2 and w and a1 and b2 and c3
can be found out respectively by using
simulative calculation methods (let calculation
step isη and number of operations is n and
Err≤Err0)
Modeling
Modeling
 If R∈[0,1] and x=R and ph=0.7 and pl=0.2 then
the normalization compression curve of
membership function of risk see Figure.
Modeling
 And then, based on the principle of maximum
degree of membership if ph=0.7 and pl=0.2 and
R<0.34 then R∈u3 and if 0.60>R≥0.34 then
R∈u2 and if R≥0.60 then R∈u1(find out R based
on formula(20), R=0.5198P + 0.4802 (1 - C)).
Or

R=0.5(P - C+1)
 p1=[0.7, 1.0], p2=[0.2, 0.7), p3=[0.0, 0.2)
 u1(R)=[0.6, 1.0], u2(R)=[0.3, 0.6),
u3(R)=[0.0, 0.3)
About hazard index(P)
P=max(pa1, pa2, pa3, …,pan)
Users can make some necessary
changes in this software.
Users can make some necessary
changes in this software.
Users can make some necessary
changes in the software ASREOH
About C
 The C consist of “weekly consumption of
hazard factor” and “level of engineering controls”
and “level of personal protective equipments”
and “level of administrative controls” etc., of
which 4 items in the drop-down menus of this
software with 5 sub-items which are in influence
degree on the risk order and the middle subitem is the general condition and then the next
is better and the previous is worse. Users can
make some necessary changes in the software
ASREOH.
About C
 C=w1c1+w2c2+w3c3…wncn
 Calculating wi by Analytic Hierarchy Process
(AHP).
Thank You