An online retailer carried out a random survey on shipments

An online retailer carried out a random survey on shipments done by its Order Fulfillment Department. By contacting customers of 3000 shipments, the retailer could find out if there are problems with the shipments. The survey response rate was 80%. Twenty one shipments were found with problems, and each of those shipments is considered an error. Find: DPU; EPMO; and the calculated sigma level of the fulfillment service. To achieve a calculated sigma level of 4.0, what would have been the number of errors? Assume that the fulfillment service is actually at 5.0 sigma. What would have been the theoretical number of errors? Can this number of errors happen? This is a continuation of part (c). What is the lowest possible number of errors? If this lowest number of errors does happen, what would be the calculated sigma level? Would this calculated sigma level be a wrong indication of the underlying sigma level? Theoretically, we would need a sample of 1 million shipments to have an absolutely accurate calculated sigma level. Given the response rate, how many customers would the retailer need to contact? Would it be costly to carry out that random survey?

Solution

a) Survey response rate = 80% then No. of observation = 3000* 0.8 = 2400

It is given that 21 shipments have problem in the above given observation

To calculate DPU = No. of defects in sample / No. of sample = 21 / 2400 = 0.00875

=> There are 0.00875 Defects per Unit.

Consider each error is same then no. of opportunities to make mistake = 1.

Now DPM = Defects per Unit X 1000000 = 0.00875 * 1000000 = 8750

=> DPMO = DPM / No. of opportunities = 8750/1 = 8750

=> DPMO value must lie between 3.0 and 4.0 Sigma level. Therefore it can be calculated as:

(4.0 - 3.0) / (6209.7-66807.2) = (X - 3.0) / (8750-66807.2)

=> 1 / 60597.5 = (X - 3.0) / 58057.2

=> X - 3.0 = 58057.2 / 60597.5 = 0.95

=> X = 3.95

=> Sigma level is 3.95

b) For Sigma level 4.0, DPMO = 6209.7

DPMO = Defects per unit * 1000000

=> Defects per unit = DPMO / 1000000 = 0.0062097

No. of defects = 0.0062097 * 2400 = 14.90 Errors.

c) Similarly calculate the Defects per unit for Sigma Level 5.0

DPU = DPMO / 1000000 = 232.6 / 1000000 = 0.0002326

Therefore no. of errors = 0.0002326 X 2400 = 0.55

=> no. of errors = 0.55 < 1 error in 2400 shipments.

Therefore it can not happen

d) In this case, Lowest possible error can be 1, i.e.1 error per 2400 shipments.

=> DPU = 1 / 2400 = 0.000416

=> DPMO = 0.000416 * 1000000 = 416

=> DPMO value must lie between 4.0 and 5.0 Sigma level. Therefore it can be calculated as:

(5.0 - 4.0) / (232.6 - 6209.7) = (X - 4.0) / (416 - 6209.7)

=> 1 / 5977.1 = (X - 4.0) / 5793.3

=> X - 4.0 = 5793.3 / 5977.1 = 0.97

=> X = 4.97

=> Sigma level is 4.97