Solved Problem on Simple probabilistic model (Individual replacement or Group replacement) -2
Example problem on Group replacement - 2
Electronic
equipment contains 1,000 resistors. When any resistor fails, it is replaced.
The cost of replacing an individual failed resistor is Rs.7. If all the
transistors are replaced simultaneously it would cost Rs. 4 per transistor. The
percent of surviving, S(i) at the end
of month i is tabulated now. Find the
optimum replacement policy (i.e) individual replacement policy or group
replacement policy.
End
of month i
|
0
|
1
|
2
|
3
|
4
|
5
|
6
|
Percent
of surviving S(i)
|
100
|
90
|
85
|
65
|
35
|
15
|
0
|
Given
Total
no of resistors = 1000
Failure
rate in first month, p1 = (100 – 90) / 100 = 0.10
Failure
rate in second month, p2 = (90 - 85) / 100 = 0.05
Failure
rate in third month, p3 = (85 - 65) / 100 = 0.20
Failure
rate in fourth month, p4 = (65 - 35) / 100 = 0.30
Failure
rate in fifth month, p5 = (35 - 15) / 100 = 0.20
Failure
rate in sixth month, p6 = (15 - 0) / 100 = 0.15
Solution
Cost of individual
replacement
Expected
life of each resistor =
=
(1*0.10) + (2*0.05) + (3*0.20) + (4*0.30) +
(5*0.20) + (6*0.15)
=
3.9 weeks
Avg.
no of failures per month = 1000 /
3.9
= 256.41
= 256 (approx.)
Cost
of individual replacement = 256 * 7
=
₹. 1792
The cost of group
replacement:
No
of resistors replaced at the end of each week listed in the following table.
Cost of group replacement of 1000 resistors = 1000*4 = ₹. 4000
End
of month
|
Total
no of resistors replaced on each month
|
Total
no of resistors replaced at the end of each month
|
Cost
of replacement of resistors
(in
₹)
|
Total
Cost
(in
₹)
|
Average
cost per month
|
|
A
|
B
|
C
|
D
= C*7
|
E
= 4000+D
|
F
= E/A
|
|
1
|
100
|
100 =
|
100
|
700
|
4700
|
4700
|
2
|
60
|
100+60 =
|
160
|
1120
|
5120
|
2560
|
3
|
211
|
211+100+60 =
|
371
|
2597
|
6597
|
2199
|
4
|
344.1
|
344+211+100+60 =
|
715
|
5006
|
9006
|
2252
|
Therefore,
the transistors replaced (group replacement) at the end of 3rd month.
Result
Here the Group replacement
cost per month is higher than the individual replacement per month. Therefore individual
replacement is recommended.
