Chapter 2 Modelling Quality and Warranty Cost - Mathematics in ...

xHUÀ¼¦½ SLF7JOvL JOw#LMJ xHH‚º¦¿
³j»F£JOvLMJTw#LMJ
JMxHVUÀ¼¿
³j»F7JTwLMJOw#L
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opt
36 CHAPTER 2. MODELLING QUALITY AND WARRANTY COST
25
Hardware Warranty Costs v/s q2, q3=0.8
10
Hardware Wararnty Costs v/s q2, q1=0.3
30
Hardware Warranty Costs v/s q1, q2=0.3
20
9.5
25
9
20
15
8.5
15
10
8
5
q1=0.1
q1=0.5
q1=0.8
7.5
7
q3=0.1
q3=0.5
q3=0.8
10
5
q3=0.1
q3=0.5
q3=0.8
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Infant Mortality Factor q2
6.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Infant Mortality Factor q2
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Component Failure Rate q1
(a) (b) (c)
Figure 2.6: (a) Trends J in hw with component JMK quality and JON IMF (JOP , fixed). (b) Trends J in hw
with JTN IMF and diagnostics JOP capability (JMK , fixed). (c) Trends J in hw with component JMK quality and
diagnostics JOP capability (JON , fixed).
2.5.2 The Software Warranty Costs
The software warranty costs are characterized in terms of the severity levels of the software failures,
SL (see Section 2.3.2). Therefore, given the operating range of the quality vector D , we can calculate
from our SL model the probabilities Êv , Ê)w , ÊŽx , Êy of failures of SL HZ_ (catastrophic) through SL H®{
(minor) respectively (see Table 2.1). Associated with each of these types of failures is a warranty cost,
Ëv , Ëw , ˆx and Ëy respectively. The warranty cost for software is expressed by:
J sw FDMHVH J FO_e…JOvH_F¯Êv>ËvVnÔÊw.ËwVnÕÊŽx7ˆxÐnÕÊ)y>Ëy_H.Q (2.10)
The SDE most significantly impacts the warranty cost of the software. A higher SDE means fewer
catastrophic failures. The probabilities Êic , º„H|lLMQMQMQL ² are fixed for a given operational range of JOv , JOw ,
, and are chosen according to our model of the Severity Level function (2.7). Recall that JTy and JOS
JMx
are given by expression (2.6). More precisely,
n…סc
ÊicŒÖ~gdH
¼ÅYRH F£]Q¯²L_5HÙØ F']Q{)L ]Q¯ÚHÍØ F']Q¯ÚL_+H where . The וc quantities are introduced to account for other
lower-order effects, which are not accounted for during the production. These include effects such as
a product being returned due to incorrect usage by the customer. One may also use a model allowing
more flexibility, such as (2.9). Unfortunately, in such a model the probabilities will need to be
computed as functions JOc of .
2.6 The Complete Model
We now summarize the models of the previous sections. Recall that the goal was to identify the various
functions in the optimization problem (2.1) repeated here for convenience