Question

Gadgets&Co sells refrigerators. Any refrigerator that malfunctions within 3 years of purchase is replaced with a new one for free. Of all refrigerators, 3% fail during their first year of operation; 5% of the one-year-old refrigerators fail within their second year of operation, and 7% of the two-year-old refrigerators fail within their 3rd year of operation.

a) Estimate analytically the fraction of all refrigerators that will have to be replaced under the 3-year warranty scheme.

b) Construct and execute a simulation model (with 120 pseudo-realities) to estimate the fraction of all refrigerators that will have to be replaced under the 3-year warranty scheme. Explain and motivate your approach.

c) Assume that it costs $650 to replace a refrigerator, and Gadgets&Co sells 15,000 refrigerators per year. Estimate analytically the replacement cost savings per year that Gadgets&Co would achieve if they decreased their warranty period to 2 years. Explain and motivate your approach and results.

d) Use the results from part b to find a simulation-based estimate of the replacement cost savings per year that Gadgets&Co would achieve if they decreased their warranty period to 2 years. Explain and motivate your approach.

Answer 1

Reasoning and understanding for all the below cases:-

When a unit in sold in Year 0, 3% is replaced in year 0, so only 97% of the original units sold become two year old in the second year. and 5% of these 97% are then replaced coming out to 4.85% of original units sold. Now in second year(3%), the units replaced in year 1 become one year old and 3% of these is to be replaced coming out to 3% of 3% = 0.09% of the original units sold. Now in third year only 100% - 3% - 4.85% = 92.15% originally sold and not replaced are 2 years old and 7% of these are replaced, coming out to 6.45% of original units sold. Units replaced in Year 1 = 3% - 0.09% = 2.91% become 1 year old and 5% of these = 5% of 2.91% = 0.145% is now replaced in the third year. Similarly, units replaced in second year,4.85% + 0.09% become one year old and 3% of these is replaced , which comes out to 0.1482% of the original units sold

a) A total of 14.54% of all units sold would be required to be replaced under the 3 year warranty scheme.

Please note that units replaced in first year will be one year old units in the second year of warranty. Accordingly on the same logic all the calculation have been made. The table and formula table are presented here.

b)

c)

d)