Question

A plant is to be built to produce blasting devices for construction work, and the decision must be made as to the extent of automation in the plant. Additional automatic equipment increases the investment costs but lowers the probability of shipping a defective device to the field, which must then be shipped back to the factory and dismantled at cost of $10 per device. The operating costs are identical for the different levels of automation. It is estimated that the plant will operate 10 years. The interest rate is 20%, and the rate of production is 100,000 devices per year for all levels of automation. Prepare a table with a column for all possible levels of automation, a column for the expected number of defectives in 100,000 devices, a column for the expected annual cost of defectives, a column for the annual cost of investment and a column for the total expected annual cost. Using the table, find the level of automation that will minimize the expected total annual cost for the investment costs and the probabilities given below

Level of Automation, Probability of producing a defective, Cost of investment ($)
1 0.100 100,000
2 0.050 150,000
3 0.020 200,000
4 0.010 275,000
5 0.005 325,000
6 0.002 350,000
7 0.001 400,000

Answer 1

As the probability of an item being defective is very less and the number of samples is huge, we use poisson distribution in this case.

For a poisson distribution the expectation is

So for each level we can calculate the expected number of defective products and hence their cost.

Here, n = 100,000 and probabilty of producing defective product is p.

For calculating annual cost of investment, we use the excel function PMT (payment), where

rate = 20 percent = 0.2

nper = 10

pv = present value of investment in the machine

optional : fv = 0 (as the plant is expected to run only for 10 years, we assume after 10 years the value of the machine will become zero)

Level	Probability of defective product	Expected number of defectives	Annual Cost of defectives(in $)	Annual Cost of Investment(in $)	Total Annual Cost(in $)
1	0.1	10,000	100,000	23,852.28	123,852.28
2	0.05	5000	50,000	35,778.41	85,778.41
3	0.02	2000	20,000	47,704.55	67,704.55
4	0.01	1000	10,000	65,593.76	75,593.76
5	0.005	500	5000	77,519.90	82,519.90
6	0.002	200	2000	83,482.96	85,482.96
7	0.001	100	1000	95,409.10	96,409.10

So, from the table we see that the minimum total annual cost is for the level 3 automation.