Group Replacement Of Items
Group Replacement of Items (cont.):
Let Ni= Number of bulbs replaced at the end of i Th month, then we can calculate different values of Ni
N 0= Number of bulbs at the beginning = 10000
N1= Number of bulbs replaced at end of first month = Number of bulbs at the beginning × Probability that a bulb fails during 1st month of installation = 10000 × 0.03 = 300
N2= Number of bulbs to be replaced at the end of second month = (Number of bulbs at the beginning × probability of failure during the second month) (Number of bulbs replaced at the end of second month × Probability of failure during the second month) = N0P2 N 1P 1= (10000 × 0.07) (300 × 0.03) = 709. Similarly,
N 3= N 0 P3 N 1 P 2 N2 P1 = 10000 × 0.20 300 × 0.07 709 × 0.03 = 2042.
N 4= N 0 P4 N1 P3 N2 P2 N 3 P1 = 10000 × 0.40 300 × 0.20 = 709 × 0.07 2042 × 0.03 = 4171.
N 5= N0 P5 N 1 P4 N 2 P3 N3 P2 N 4 P1 = 10000 × 0.15 300 × 0.40 709 × 0.20 2042 × 0.07 4171 × 0.03 = 2030.
N6= N1 P5 N2 P4 N3 P3 N4 P2 N5 P1 = 10000 × 0.15 300 × 0.15 709 × 0.40 2042 × 0.20 4171 × 0.07 2030 × 0.03 = 2590.
From the above we can see that the failures increase from 300 to 4171 in 4th month and then decreases.
It is very much common in any system that failure rate increases and then decreases and finally after certain period, it stabilizes, when the system attains steady state.
Now work out the expected life of each bulb which is = Σ xi Pi, where xi is the month and Pi is corresponding probability of failure.
Σ xi Pi= 1 × 0.03 2 × 0.07 3 × 0.20 × 4 × 0.40 5 × 0.15 6 × 0.15 = 4.02 months.
If the average life of a bulb is 4.02 months, the average number of replacements every month =Number of bulbs in the system / average life of the bulb. = 10000 / 4.02 = 2488 bulbs.
As the cost ofindividual replacement cost is Re.1/- per bulb, on an average, the organization has to spend 2488 ×Re.1/- = Rs. 2488/- per month
In the above table in column No. 2 it is shown the cost of individual replacement at the end of month plus group replacement of all the bulbs at the end of month.
The minimum cost of group replacement i.e. Rs. 2183.66 is at the end of third month. This is compared with individual replacement cost per month, which is Rs. 2488/- per month. Hence replacement of all the bulbs at the end of third month is more beneficial to the organization. Hence optimal replacement policy is replacing all the bulbs at the end of third month.