Question

vehicle transmission repairs for part number tac45123 are costly to replace. a highly skilled mechanic must disassemble the transmission. if the part fails while it is in warranty, the manufacturer pays the full cost of the repair to the garage or dealer that did the repair. the manufacturer uses the taguchi loss function to set a specification for the part with k = 30,000. the dimensions for the part are 1.0 ± 0.12 centimeters. round your answers to the nearest dollar.
a. what is the economic value of one failed part?
$

b. if the manufacturer of the part paid the full warranty cost on 1,896 failures last year worldwide, what is the total economic cost of failure?
$

c. if the manufacturer improved process and equipment capability so the part specifications changed to 1.0 ± 0.06 centimeters, what is the economic cost of one failure?
$

d. assuming failures were reduced by one half to 948 due to the more precise specifications discussed in part c, what is the economic cost of failure?
$

e. how much warranty cost is saved according to taguchi estimates by improving specifications from ±0.12 to ±0.06?
$

Explanation:

Part a

Step1: Recall Taguchi loss function

The Taguchi loss function is given by \( L(x) = k(x - T)^2 \), where \( L(x) \) is the loss, \( k \) is the loss coefficient, \( x \) is the actual value, and \( T \) is the target value. For a failed part, the deviation from the target is at the specification limit. The target \( T = 1.0 \) cm, and the specification limit is \( \pm 0.12 \) cm, so \( x - T = 0.12 \) (or -0.12, but squared so it's the same).

Step2: Calculate loss for one failed part

Substitute \( k = 30000 \) and \( (x - T) = 0.12 \) into the formula: \( L = 30000\times(0.12)^2 \)
\( L = 30000\times0.0144 = 432 \)

Step1: Multiply cost per failure by number of failures

We know from part a that the cost per failed part is \( \$432 \), and the number of failures is \( 1896 \). So total cost is \( 432\times1896 \)

Step2: Calculate the product

\( 432\times1896 = 432\times(1900 - 4)=432\times1900 - 432\times4 = 820800 - 1728 = 819072 \)

Step1: Use Taguchi loss function with new specification

Now the specification limit is \( \pm 0.06 \) cm, so \( (x - T)=0.06 \). The loss function is still \( L(x)=k(x - T)^2 \) with \( k = 30000 \)

Step2: Calculate the loss

\( L = 30000\times(0.06)^2=30000\times0.0036 = 108 \)

Answer:

\( 432 \)

Part b