Question

The Acme Manufacturing Company employs 500 machine operators. These operators are classified as apprentice, machinist, and master machinist. Company policy dictates that all new operators be hired at the apprentice level. Currently, there are 200 apprentices, 250 machinists, and 50 master machinists. Acme’s union contract specifies that of the employees that start the new year at the company, 15% in each class are to be promoted each year. It is estimated that an additional 15% of the employees that started as apprentices leave the company each year, 10% of the machinists leave, and 5% of the master machinists leave. Due to current economic conditions, the company does not rehire and replace these.

Define a Markov chain to model this situation, and construct its one-step transition matrix.

Answer 1

The one-step transition matrix is:

`\left[\begin{array}{cccc}0.70&0.15&0&0.15\\0&0.90&0&0.10\\0&0&0.95&0.05\\0&0&0&1\end{array}\right]`

Explanation:

We have 4 states for a the employees:

1. Apprentice (A)
2. Machinist (M)
3. Mastet machinist (MM)
4. Out of the company (O)

Of the apprentices (A), 15% are promoted each year to machinists (M) and 15% leave the company (O). Then, 60% of the apprentices stay in the same job.

Of the machinists (M), 10% leave the company (O), so 90% stay as machinists.

Of the master machinists (MM), 5% leave the company (O), so 95% stay as master machinists.

The transition matrix can be written as:

`\left|\begin{array}{ccccc}&A&M&MM&O\\A&0.70&0.15&0&0.15\\M&0&0.90&0&0.10\\MM&0&0&0.95&0.05\\O&0&0&0&1\end{array}\right|`

We can calculate the future step from the actual state as:

`[\begin{array}{cccc}200 &250& 50& 0\end{array}]*\left[\begin{array}{cccc}0.70&0.15&0&0.15\\0&0.90&0&0.10\\0&0&0.95&0.05\\0&0&0&1\end{array}\right]=[\begin{array}{cccc}140 &255& 47.5& 57.5\end{array}]`