Grease, Inc. is a producer of a variety of machine parts. The company has recently purchased an automatic machining center to economically produce a new product GT100. The machining center operates as follows: There are two machining stages involved in the production of GT100s. The machining center performs both stages. After the first stage, an automatic inspection takes place. Depending on the result of the inspection, either stage 1 is repeated, the product is scrapped (with a probability of 0.05), or stage 2 is undertaken (with a probability of 0.85). 

After stage 2 is completed, the product is unloaded from the machine and a new raw material stock is loaded for machining (which will begin with stage 1). Similarly, if a product is scrapped after stage 1, it is removed from the machine, a new raw material stock is loaded, and the machining cycle begins from stage 1. Both machining stages take the same amount of time (60 seconds). The inspection time after stage 1 is negligible. The operation of unloading a product (finished or defective) from the machine and loading a new raw material stock takes 1 minute. 

(a) Model this situation as a Markov chain. 

(b) Determine the steady-state probabilities. 

(c) Determine the long-run average hourly production rate of the machining center (i.e., the long-run average number of produced products per hour). 

(d) The raw material needed for producing one GT100 costs $10. This raw material cost cannot be recovered for scrapped products. The selling price per unit of GT100 is set at $20, and the hourly cost of operating the machining center is $50. Determine the long-run average hourly profit from the sale of GT100s.

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Solution Modeling Grease Incs Production Process as a Markov Chain a Modeling the situation as a Mar... View the full answer