A manufacturing process requires three operations, A, B and
C, to be completed sequentially. A worker takes a raw material
kit and begins operation A. The operation takes one week. At
the end of the week an inspector checks the operation. If it
is successful, the worker begins operation B. If it is not successful,
the worker repeats operation A. Because of quality considerations,
operation A can be tried no more than two times. If the inspection
fails on the second attempt, the product is discarded and the
worker starts operation A using a new raw material kit. If it
is successful, the worker begins operation B. The probability
of a successful inspection is the same on the first and second
tries and it is shown in the table below.
Operations B and C are similar to operation A, but have different
success probabilities. Each attempt to complete an operation
takes one week, and only two tries are allowed. When operation
B is completed successfully, the worker starts operation C.
If operation B is not completed successfully on the second try,
the product is discarded and the worker starts operation A using
a new raw material kit. When operation C is completed successfully,
the product is sold. If operation C is not completed in two
tries, the product is discarded. In either case, the worker
then begins at operation A with a new raw material kit.
Answer numerical questions with the Stochastic Analysis Add-in.
a. Construct the DTMC Matrix that describes this manufacturing
b. What is the steady-state production rate of this system?
c. What proportion of the time does the worker spend on each
of the operations?
d. What proportion of the kits started will be completed successfully.
e. At the beginning of January the worker begins a new kit
with operation A. What is the probability distribution of the
number of kits completed after 12 weeks?
f. Say a raw material kit for one unit of product has a cost
of $500, and that the selling price for a completed unit is
$2000. The worker's labor cost is $200 per week. Estimate the
profit for 52 weeks of operation.