MHB Optimizing Markov Chain Production System Throughput w/ Exponential Rate of 50

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The discussion focuses on optimizing the throughput of a wind turbine manufacturer's production system by installing a buffer between pre-assembly and final assembly. The manufacturer aims to balance the costs of buffer spaces, which are €5,000 per month, against the profit of €10,000 per turbine produced. The production times for both assembly stages are exponentially distributed with a rate of 50, and the absence of failures simplifies the initial modeling. The conversation explores the trade-offs between buffer space costs and production efficiency, highlighting the need for a clear model to determine the optimal number of buffer spaces. The framing of the problem suggests a complex interplay between production rates and buffer management.
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I am new to Markov chain, i want to model this as a continuous-time Markov chain.

A wind turbine manufacturer would like to increase the throughput of its production system. For this purpose it intends to install a buffer between the pre-assembly and the final assembly of the wind turbines. The manufacturer can generate a profi t of 10.000 Euro per wind turbine. However, buffer spaces are also fairly expensive. The company estimates that one buffer space costs 5000 Euro per month. The production times for one turbine in the pre-assembly and in the final assembly are both exponentially distributed with a rate of 50. Again we assume that no failures occur, neither in the pre-assembly nor in the final production.

a) What is the optimal amount of buffer spaces the company should install? Determine the corresponding monthly pro ts.

b) Imagine the company realizes that failures can occur in both, prea-ssembly and fi nal production. How would you model this production system?

The transition(production) rate of moving from a pre-assembly to final assembly is 50.

I really don`t know how to model this problem.
 
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Hi there,

Welcome to MHB!

I'm trying to think through this problem as it's not immediately clear to me but I have a hunch. It sounds like we have two exponential processes and this extra part is a bridge between the two. The best case would be that there is no waiting time in the buffer for the $(n-1)$ iteration to complete in the final-assembly. The worst case would be that all the buffer spaces are filled and that the pre-assembly needs to stop in order for the final-assembly to complete the $(n-1)$ iteration and for another buffer space to open up. The trade off between the two is the cost of the buffer.

Does that sound like a correct framing of the process?
 

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