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You are part of the Operations Team at Gamma motors, investigating the painting process. The painting machine naturally requires an average of 10 hours to complete with a standard deviation of 2 hours. After some observation, you determine that the process seems to have a lot of breakdowns. Based on the data you collected, the average time between failures of the painting machine appears to be a mere 200 hours with a standard deviation of 50 hours. Repair times after a failure average 10 hours with a standard deviation of 10 hours. The factory only has one painting machine, which is the only machine at the workstation.

a. Identify the Effective Process Time (te):
b. SCV of Effective Process times (Ce2):
c. Standard Deviation of Effective Process times (Sigmae):

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Answer:

Part a: The effective time is 10.5 hours.

Part b: The SCV of Effective Process times is 0.130.

Part c: The value of the Standard Deviation of the Effective process times is 3.80

Explanation:

Part a:

The effective time t_e is given as

[tex]t_e=\dfrac{t_p}{A}[/tex]

Here t_p is the time taken to complete the paint job which is given as 10

A is the availability which is given as

[tex]A=\dfrac{M_{TBF}}{M_{TBF}+M_R}[/tex]

here M_TBF is the mean time between failures which is given as 200 hours

M_R is the mean repair time which is given as 10 hours so

[tex]A=\dfrac{M_{TBF}}{M_{TBF}+M_R}\\A=\dfrac{200}{200+10}\\A=\dfrac{200}{210}=0.9524[/tex]

So the effective time is calculated as

[tex]t_e=\dfrac{t_p}{A}\\t_e=\dfrac{10}{0.9524}\\t_e=10.499\approx 10.5\\[/tex]

So the effective time is 10.5 hours.

Part c:

The Standard Deviation of the Effective process times is given as

[tex]\sigma_e^2=\left(\dfrac{\sigma_p}{A}\right)^2+\dfrac{(M_R^2+\sigma_R^2)(1-A)t_p}{AM_R}[/tex]

Here

σ_e is the standard deviation of the effective process time which is to be calculated

σ_p is the standard deviation of time taken to complete the paint job given as 2

A is the availability calculated above as 0.9524

M_R is the mean time of repair given as 10

σ_R is the standard deviation of time taken to repair given as 10

t_p is the time taken to complete a paint job which is given as 10

so the equation becomes

[tex]\sigma_e^2=\left(\dfrac{\sigma_p}{A}\right)^2+\dfrac{(M_R^2+\sigma_R^2)(1-A)t_p}{AM_R}\\\sigma_e^2=\left(\dfrac{2}{0.9524}\right)^2+\dfrac{(10^2+10^2)(1-0.9524)*10}{0.9524*10}\\\sigma_e^2=4.4098+9.9958\\\sigma_e^2=14.4056\\\sigma_e=\sqrt{14.4056}\\\sigma_e=3.7954[/tex]

So the value of the Standard Deviation of the Effective process times is 3.80

Part b

The SCV of Effective Process times is given as

[tex]C_e^2=\dfrac{\sigma_e^2}{t_e^2}[/tex]

here both values of σ_e and t_e are calculated in above parts as 3.89 and 10.5 respectively so the equation becomes.

[tex]C_e^2=\dfrac{\sigma_e^2}{t_e^2}\\C_e^2=\dfrac{3.80^2}{10.5^2}\\C_e^2=0.130[/tex]

So the SCV of Effective Process times is 0.130

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