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 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 onestep transition matrix.

Answer 1

Based on the situation, we model it as a Markov chain with four states - apprentice (State A), machinist (State M), master machinist (State MM) and left the company. (State L) Currently, there are 200 apprentices, 250 machinists, and 50 master machinists. So, the initial probability vector is (200/500, 250/500, 50/250, 0]

= [0.4, 0.5, 0.1, 0]

15 % in each class are to be promoted each year. Hence the transition probability from State A to State M is 0.15

The transition probability from State M to State MM is 0.15

Additional 15 % of the employees that started as apprentices leave the company each year. So, the transition probability from State A to State L is 0.15. The transition probability from State A to State A is 1 - (0.15 + 0.15) = 0.7

Additional 10 % of the machinists leave the company each year. So, the transition probability from State M to State L is 0.10. The transition probability from State M to State M is 1 - (0.15 + 0.10) = 0.75

Additional 5% of the master machinists leave the company each year. So, the transition probability from State MM to State L is 0.05. The transition probability from State MM to State MM is 1 - 0.05 = 0.95

Due to current economic conditions, the company does not rehire and replace these. So, the transition probability from State L to any of the states A, M or MM is 0. The transition probability from State L to State L is 1.

The one step transition matrix with the states in order A, M, MM, L is,