Problems CTMC Models
 Continuous Time Markov Chain Models - Serial Production System

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.

 Operation A B C Success Probability 0.90 0.95 0.85

a. Construct the CTMC 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. 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 discarded?

Operations Research Models and Methods
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by Paul A. Jensen