The following table gives probability distribution of failures observed for a certain type of light bulbs:

There are 10,000 bulbs in use and it costs Rs. 20 to replace an individual bulb which has burnt out at the end of the week. If all bulbs are replaced simultaneously it would cost Rs. 14 per bulb. It is proposed to replace all bulbs at fixed interval of time, irrespective of their status and to continue replacing burnt out bulbs at the end of every week. At what interval should all the bulbs be replaced? What is the average cost of replacement if all the bulbs are replaced as per individual replacement policy? What is the optimal replacement policy?

Mumbai University > Mechanical Engineering > Sem 7 > Operations Research

Marks: 10 Marks

Year: Dec 2015

Let $p_i$ be the probability that a new light bulb fails during the ith week of its life. Then we get following probability distribution of the lives of the bulb:

$p_1=0.05 \\ p_2=0.10 \\ p_3=0.25 \\ p_4=0.35 \\ p_5=0.25$

Since the sum of all probabilities is unity, all probabilities higher than p5 is 0.

Thus all the light bulbs burnt out by the end of 5th week.

From this distribution that 5 percent of bulbs are expected to burn out during the first week of their life. In the lot of 10000 bulbs, therefore, 500 bulbs are expected to fail in the first week. Similarly of this lot 10% i.e., 1000 bulbs are expected to burnt out at the end of second week.,2500 bulbs in third week 3500 bubs in fourth week, 2500 in fifth week.

The number of failures (and hence number of replacements) in different weeks are calculated in table below:

Thus we find that the number of bulbs failing each week increases till 3rd week then decreases.

1.Determination of optimal group replacement interval:

As we can see from above table the cost up to 3rd week is lowest therefore it is optimal to have a group replacement after every two weeks.

Average cost of replacement is Rs. 74173.33 for all the bulbs as per group replacement policy.

2.When the policy of replacing the bulbs as and when they fail is adopted there are no group replacement and it becomes an individual replacement policy/ since it’s assume that bulb can fail at any time in the week it is necessary to first determine average life of bulb.

Average (expected) life of light bulbs $$= \sum_{i=1}^6 ip_i \\ =1 \times 0.05+2 \times 0.10+3 \times 0.25+4 \times 0.35+5 \times 0.25=3.65$$

Average number of failure per week = $\dfrac{10000}{3.65}=2739.7$

Therefore cost of individual replacement of bulbs (per week) = 2739.7 x 20 =54794.52

As the cost of group replacement is 74173.33 and cost of individual replacement is 54794.52 its advisable to adopt the individual replacement policy.