Max$$s B$-$B$-$Q Inc. manufactures top$-$of$-$the$-$line barbeque tools. The tools include forks, spatulas, knives, spoons, and shish$-$kebab skewers. Max$$s fabricates both the metal parts of the tools and the resin handles. These are then riveted together to create the tools $($Figure C$9\mathrm{.}2\mathrm{.}1).$ Recently, Max$$s hired you as a process engineer. Your first assignment is to study routine tool wear on the company$$s stamping machine. In particular, you will be studying tool$-$wear patterns for the tools used to create knife blades.n the stamping process, the tooling wears slightly during each stroke of the press as the punch shears through the material. As the tool wears, the part features b ecome smaller. The knife has specifications of$10$ mm $+-0\mathrm{.}025$ mm undersized parts must be scrapped. The tool can be resharpened to bring the parts produced back into specification. To reduce manufacturing costs and simplify machine scheduling, it is critical to pull the tool and perform maintenance only when absolutely necessary. It is very important for scheduling, costing, and quality purposes that the average number of strokes, or tool run length, be determined. Knowing the average number of strokes that can be performed by a tool enables routine maintenance to be scheduled.It is the plant manager$$s philosophy that tool maintenance be scheduled proactively. When a tool is pulled unexpectedly, the tool maintenance area may not have time to work on it immediately. Presses without tools don$$t run, and if they are not running, they are not making money. As the process engineer studying tool wear, you must develop a prediction for when the tool should be pulled and resharpened.The following information is available from the tool maintenance department.

$$The average number of strokes for a tool is $45,000.$

$$The standard deviation is $2,500$ strokes.

$$A punch has a total of $25$ mm that can be ground off before it is no longer useful.

$$Each regrind to sharpen a punch removes $1$ mm of punch life.

$$The cost to regrind is

$2$ hours of press downtime to remove and reinsert tool, at $$300$ per hour

$5$ hours of tool maintenance time, at $$65$ per hour

$5$ hours of downtime while press is not being used, at $$300$ per hour

$$The average wait time for unplanned tool regrind is $15$ hours at $$300$ per hour.

$$Because of the large number of strokes per tool regrind, this is considered to be a continuous distribution. The normal curve probability distribution is applicable.AssignmentOne percent of the tools wear out very early in their expected productive life. Early tool wearout$$and$,$ thus, an unplanned tool pull$$can be caused by a variety of factors, including changes in the hardness of the material being punched, lack of lubrication, the hardness of the tool steel, and the width of the gap between the punch and the die. Key part dimensions are monitored using and R charts. These charts reveal when the tool needs to be reground in order to preserve part quality. Use the normal probability distribution and the information provided to calculate the number of strokes that would result in an early wearout percentage of $1$ percent or fewer. If the plant manager wants the tool to be pulled for a regrind at $40,000$ strokes, what is the chance that there will be an early tool wearout failure before the tool reaches $40,000$ strokes?