Problems DTMC Models
 Discrete Time Markov Chain Models - Serial Production

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.

 Operation A B C Success Probability 0.90 0.95 0.85

Answer numerical questions with the Stochastic Analysis Add-in.

a. Construct the DTMC Matrix that describes this manufacturing process.

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.

Operations Research Models and Methods
Internet
by Paul A. Jensen