A plastic manufacturing company produces PVC pipes for a customer who sets the specification of the length of the pipes at 125 mm with a +/- 10% tolerance. The customer has set the quality standard of the delivery batch of the products that allows a maximum of 6% of defective products (i.e. the products length falls outside the specification limits) to accept the whole batch delivered. The manufacturer uses a processing machine to produce the PVC pipes with a capacity of 5,000 units produced within the 8 hours shift per day. The record of the control chart shows that the machines outputs have an average length of 129.5 mm with a standard deviation of 6 mm. The customer places a daily order of 5,000 units of PVC pipes, and the factory is determined to fulfill these orders consistently. The cost of material is $0.10 per mm.

1. Calculate the percentage of defects that the machine will produce against the specification of the product. Can the factorys outputs that are produced within the normal shift meet the quality standard set by the customers?

2. What possible solution/s you can offer to the company to meet the quality standard set by the customer without changing the setting or condition of the machine? Show some calculations to support your solution and the implications of your solution

3. If the factory can adjust the setting of the machine to change the average length of its outputs (without changing the standard deviation), determine the setting that the manufacturer should choose to minimize the production cost for producing the PVC pipes but still meets the quality standard set by the customer. Show the calculations and explain your answers.