Advanced Mathematics and Numerical Modeling of IoT

Table 3: Possible combination of the risk factors and weights.

GRPN

(퐷푖,푇푖)

Combination of (푤푆,푤푂,푤퐷), GRPN

TRPN

(L, M, H) (L, H, M) (M, L, H) (M, H, L) (H, L, M) (H, M, L) (E, E, E)

(푆,푂,퐷)

(UUU) (퐷 1 ,푇 1 )(퐷 2 ,푇 2 )(퐷 3 ,푇 3 )(퐷 4 ,푇 4 )(퐷 5 ,푇 5 )(퐷 6 ,푇 6 )(퐷 7 ,푇 7 ) TRPN1

(UUN) (퐷 8 ,푇 8 )(퐷 9 ,푇 9 )(퐷 10 ,푇 10 )(퐷 11 ,푇 11 )(퐷 12 ,푇 12 )(퐷 13 ,푇 13 )(퐷 14 ,푇 14 ) TRPN2

(UNU) (퐷 15 ,푇 15 )(퐷 16 ,푇 16 )(퐷 17 ,푇 17 )(퐷 18 ,푇 18 )(퐷 19 ,푇 19 )(퐷 20 ,푇 20 )(퐷 21 ,푇 21 ) TRPN3

(UNN) (퐷 22 ,푇 22 )(퐷 23 ,푇 23 )(퐷 24 ,푇 24 )(퐷 25 ,푇 25 )(퐷 26 ,푇 26 )(퐷 27 ,푇 27 )(퐷 28 ,푇 28 ) TRPN4

(NUU) (퐷 29 ,푇 29 )(퐷 30 ,푇 30 )(퐷 31 ,푇 31 )(퐷 32 ,푇 32 )(퐷 33 ,푇 33 )(퐷 34 ,푇 34 )(퐷 35 ,푇 35 ) TRPN5

(NUN) (퐷 36 ,푇 36 )(퐷 37 ,푇 37 )(퐷 38 ,푇 38 )(퐷 39 ,푇 39 )(퐷 40 ,푇 40 )(퐷 41 ,푇 41 )(퐷 42 ,푇 42 ) TRPN6

(NNU) (퐷 43 ,푇 43 )(퐷 44 ,푇 44 )(퐷 45 ,푇 45 )(퐷 46 ,푇 46 )(퐷 47 ,푇 47 )(퐷 48 ,푇 48 )(퐷 49 ,푇 49 ) TRPN7

(NNN) (퐷 50 ,푇 50 )(퐷 51 ,푇 51 )(퐷 52 ,푇 52 )(퐷 53 ,푇 53 )(퐷 54 ,푇 54 )(퐷 55 ,푇 55 )(퐷 56 ,푇 56 ) TRPN8

Table 4: An example of corrective action on the failure mode (FM) with respect to푇.

Risk weight∗ GRPN Action or not with respect to given threshold푇∗∗
푤푆 푤푂 푤퐷 푇 = 0.60 푇 = 0.65 푇 = 0.70 푇 = 0.75 푇 = 0.80
0.1 0.3 0.6 0.78 Yes Yes Yes Yes No
0.1 0.6 0.3 0.72 Yes Yes Yes No No
0.3 0.1 0.6 0.70 Yes Yes Yes No No
0.3 0.6 0.1 0.60 Yes No No No No
0.6 0.1 0.3 0.52 No No No No No
0.6 0.3 0.1 0.48 No No No No No
∗(푆,푂,퐷)=(2,5,8)andGRPN=log(푆푤푆⋅푂푤푂⋅퐷푤퐷).
∗∗푇denotedasGRPNthreshold푇
푖.

(GRPN threshold) for the푖th combination. In addition, for
comparison, TRPNs (TRPN1∼TRPN8) are also simulated
with the risk factors (SOD) in both uniform and normal
distributions. The overall combination is shown inTable 3.
Risk analysis is based on an acceptable risk probability훼,
which is assigned by organization, department, or process.
Given훼, a threshold can be precalculated, where Prob
(GRPN ≤푇푖)=훼and the probability that the GRPN
value is less than or equal to푇푖is equal to훼,asshownin
Figure 1. Whenever the GRPN value>푇푖,priorityshould
begiventotakecorrectiveactiononthefailuremode
(FM). The threshold푇푖canbeanalyzedandsuggestedbya
simulation approach with respective scenarios, as discussed
inSection 3.3.
In this section, we give an example to show how the
proposed model works. Having an FM, for example, we assign
ariskfactor(푆,푂,퐷) to (2,5,8), and the risk weight (L,M,H)
is given (0.1, 0.3, 0.6). Corrective action should be taken on
theFMwheneveritsGRPNvalueisgreaterthanorequalto
agiventhreshold푇푖. The weight combination (푤푆,푤푂,푤퐷)
is illustrated inTable 4to show whether we should act on the
FM.Tousetheproposedmodel,wesummarizetheprocedure
in the following steps in which GRPN耠and GRPN耠耠denoted
the GRPN values that are calculated in simulated and real
environments, respectively. The most important thing is to
decide the threshold푇푖by a simulation process with a given
훼.

(1) Use historical data of risk factors (푆,푂,퐷)tobuilda

probability model of the factors.

GRPN

Frequency

Distribution ofDi

GRPN threshold (Ti) of

acceptable risk level

GRPN probability (1−훼)of

unacceptable risk level

GRPN probability (훼)of

acceptable risk level

Figure 1: Analysis of the acceptable risk.

(2) Give훼and risk weights (푤푆,푤푂,푤퐷) according to

organization policy.

(3) Suggest threshold푇푖by analyzing GRPN耠with prob-

ability model from step (1), such that Prob (GRPN≤

푇푖)=훼.

(4) Create an FMEA worksheet (a comprehensive work-

sheet example will be given in Section 5.1), and

compute GRPN耠耠on all FMs.

(5) Act on FMs whose GRPN耠耠are greater than푇푖.

(6) Repeat steps (2) to (5) until GRPN耠耠≤푇푖.