Solved Problem on Simple probabilistic model (Individual replacement or Group replacement) -1
Example problem on Group replacement - 1
The
failure rates of transistors in a computer are summarized in the following
table. The cost of replacing an individual failed transistor is Rs.7. If all
the transistors are replaced simultaneously it would cost Rs. 4 per transistor.
Find the optimum replacement policy (i.e) individual replacement policy or
group replacement policy.
End
of week
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
Probability
of failure
|
0.06
|
0.17
|
0.31
|
0.48
|
0.68
|
0.90
|
1.00
|
Given
Assume
that there are 100 transistors in use.
Failure
rate in first week, p1 = 0.06
Failure
rate in second week, p2 = 0.17 – 0.06 = 0.11
Failure
rate in third week, p3 = 0.31 – 0.17 = 0.14
Failure
rate in fourth week, p4 = 0.48 – 0.31 = 0.17
Failure
rate in fifth week, p5 = 0.68 – 0.48 = 0.20
Failure
rate in sixth week, p6 = 0.90 – 0.68 = 0.22
Failure
rate in seventh week, p7 = 1.00 – 0.90 = 0.10
Solution
Cost of individual
replacement
Expected
life of each transistor = (1*p1) + (2*p2) + .... + ((7*p7)
=
(1*0.06) + (2*0.11) + (3*0.14) + (4*0.17) + (5*0.20)
+ (6*0.22) + (7*0.10)
=
4.28 weeks
Avg.
no of failures per week =
100 / 4.28
= 23.36
= 23 (approx.)
Cost
of individual replacement = 23 * 7
=
₹. 207
The cost of group
replacement:
No
of transistors replaced at the end of each week listed in the following table.
Cost of group replacement of 100 transistors = 100*4 = ₹. 400
End
of Week
|
Total
no of transistors replaced on each week
|
Total
no of transistors replaced at the end of each week
|
Cost
of replacement of transistors
(in ₹)
|
Total
Cost
(in ₹)
|
Average
cost per week
(in ₹)
|
|
A
|
B
|
C
|
D =
C*7
|
E =
400+D
|
F =
E/A
|
|
1
|
6
|
6 =
|
6
|
6*7 =
42
|
442
|
442
|
2
|
12
|
6+12
=
|
18
|
18*7=
126
|
526
|
263
|
3
|
16
|
6+12+16
=
|
34
|
34*7
= 238
|
638
|
212.6
|
4
|
21
|
6+12+16+21
=
|
55
|
55*7
= 385
|
785
|
196.25
|
5
|
26
|
6+12+16+21+26
=
|
81
|
81*7
= 567
|
967
|
193.4
|
6
|
32
|
6+12+16+21+26+32
=
|
113
|
113*7
= 791
|
1191
|
198.5
|
Therefore,
the transistors replaced (group replacement) at the end of 5th week.
Result
Here the Group replacement
cost per week is less than the individual replacement per week. Therefore group
replacement is recommended.
