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 time for the operation has an
exponential distribution with a mean of one week. When the worker
finishes A, an inspector checks the operation. If it is successful,
the worker begins operation B. If it is not successful, the
worker repeats operation A. The process is repeated as many
times as necessary until it is successful. The probability of
a successful inspection is the same on each try 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
has an average time of one week. When operation B is completed
successfully, the worker starts operation C. If operation B
is not completed successfully the process repeats. The process
repeats as many times as necessary until it is successful. When
operation C is completed successfully, the product is sold.
Again, if this process is not successful it is repeated. When
process C is completed successfully, the worker immediately
begins operation A with a new raw material kit.
All processing times have exponential distributions.
Answer numerical questions with the Stochastic Analysis Add-in.
a. Construct the CTMC 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. 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?
e. 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.
f. Analyze the system when the success probabilities are all
1. Compare the profit with and without failures.
g. Change the original situation so that at most two tries
are allowed at each operation. What proportion of the kits are