1. **Problem Statement:** Miss Mshahara has data on employee movements between jobs at Edge Manufacturers Limited (EML) from 2022 to 2023. We need to analyze the transition probabilities, interpret them, find the initial total employees, and identify excess or inadequate supply in 2023. 2. **Given Data:** The movement counts from 2022 to 2023 are: $$\begin{array}{c|ccccc} & PM & FP & TL & PW & Exit \\ \hline PM & 4 & 0 & 0 & 0 & 1 \\ FP & 4 & 28 & 0 & 0 & 8 \\ TL & 0 & 12 & 96 & 12 & 0 \\ PW & 0 & 0 & 60 & 740 & 200 \\ \end{array}$$ Projected supply for 2023: PM=8, FP=40, TL=156, PW=752. 3. **i) Transition Probability Matrix:** Transition probabilities are calculated by dividing each count in a row by the total employees in that row (2022). Calculate row totals (2022 employees): - PM: $4+0+0+0+1=5$ - FP: $4+28+0+0+8=40$ - TL: $0+12+96+12+0=120$ - PW: $0+0+60+740+200=1000$ Calculate probabilities for each row: - PM row: $\left[\frac{4}{5}, \frac{0}{5}, \frac{0}{5}, \frac{0}{5}, \frac{1}{5}\right] = [0.8, 0, 0, 0, 0.2]$ - FP row: $\left[\frac{4}{40}, \frac{28}{40}, \frac{0}{40}, \frac{0}{40}, \frac{8}{40}\right] = [0.1, 0.7, 0, 0, 0.2]$ - TL row: $\left[\frac{0}{120}, \frac{12}{120}, \frac{96}{120}, \frac{12}{120}, \frac{0}{120}\right] = [0, 0.1, 0.8, 0.1, 0]$ - PW row: $\left[\frac{0}{1000}, \frac{0}{1000}, \frac{60}{1000}, \frac{740}{1000}, \frac{200}{1000}\right] = [0, 0, 0.06, 0.74, 0.2]$ Transition matrix $P$: $$ P = \begin{bmatrix} 0.8 & 0 & 0 & 0 & 0.2 \\ 0.1 & 0.7 & 0 & 0 & 0.2 \\ 0 & 0.1 & 0.8 & 0.1 & 0 \\ 0 & 0 & 0.06 & 0.74 & 0.2 \end{bmatrix} $$ **Difference from norm:** Normally, transition matrices are square and stochastic with rows summing to 1. Here, the matrix includes an "Exit" column representing leaving the company, which is not a job state but an absorbing state. This makes the matrix non-square if considering only jobs, but here it is augmented to include exit, so rows still sum to 1. 4. **ii) Meaning of the third row probabilities:** The third row corresponds to employees who were Team Leaders (TL) in 2022. - $0$ to PM: No TLs moved to PM. - $0.1$ to FP: 10% of TLs moved to Foreperson. - $0.8$ stayed TL: 80% remained Team Leaders. - $0.1$ to PW: 10% moved to Production Worker. - $0$ exited: No TLs left the company. This shows TLs mostly stay in their role, some move down or sideways, none exit. 5. **iii) Initial total number of employees (2022):** Sum all employees in 2022: $$5 + 40 + 120 + 1000 = 1165$$ 6. **iv) Jobs with excess or inadequate supply in 2023:** Compare projected supply with total employees in 2023 (sum of columns excluding Exit): Calculate 2023 totals per job: - PM: $4 + 4 + 0 + 0 = 8$ - FP: $0 + 28 + 12 + 0 = 40$ - TL: $0 + 0 + 96 + 60 = 156$ - PW: $0 + 0 + 12 + 740 = 752$ Projected supply: - PM: 8 - FP: 40 - TL: 156 - PW: 752 Since actual matches projected exactly, there is no excess or shortage. **Answer:** - a) Excess supply: None - b) Inadequate supply: None --- **Final answers:** - i) Transition matrix as above, includes exit state, rows sum to 1. - ii) Third row shows TLs mostly stay, some move to FP or PW, none exit. - iii) Initial total employees = 1165. - iv) No excess or shortage in 2023 jobs.