### Step-by-Step Solution:
#### Part (a): Labor Productivity
1. Initial Data:
- Before improvement: 13.333 carts per worker per hour
- After improvement: 16.800 carts per worker per hour
2. Calculate the increase in labor productivity:
[tex]\( \text{Increase in labor productivity} = \text{After} - \text{Before} \)[/tex]
[tex]\[ \text{Increase in labor productivity} = 16.800 - 13.333 = 3.467 \][/tex]
3. Calculate the percentage increase in labor productivity:
[tex]\[ \text{Percentage Increase} = \left( \frac{\text{Increase}}{\text{Before}} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage Increase} = \left( \frac{3.467}{13.333} \right) \times 100 \approx 26.0\% \][/tex]
#### Part (b): Multifactor Productivity
1. Initial Data:
- Before improvement: 0.635 carts per dollar cost
- After improvement: 0.694 carts per dollar cost
2. Calculate the increase in multifactor productivity:
[tex]\( \text{Increase in multifactor productivity} = \text{After} - \text{Before} \)[/tex]
[tex]\[ \text{Increase in multifactor productivity} = 0.694 - 0.635 = 0.059 \][/tex]
3. Calculate the percentage increase in multifactor productivity:
[tex]\[ \text{Percentage Increase} = \left( \frac{\text{Increase}}{\text{Before}} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage Increase} = \left( \frac{0.059}{0.635} \right) \times 100 \approx 9.29\% \][/tex]
#### Part (c): Comment on Changes in Productivity
- Labor Productivity:
- The labor productivity increased by 3.467 carts per worker per hour, which is an increase of approximately 26.0%.
- Multifactor Productivity:
- The multifactor productivity increased by 0.059 carts per dollar cost, which is an increase of approximately 9.29%.
The increase in both labor productivity and multifactor productivity indicates improvement in overall efficiency. Labor productivity showed a significant increase, suggesting that workers are producing more carts per hour of work. Multifactor productivity also saw an increase, albeit a smaller one, indicating that the overall effectiveness of using both labor and equipment has improved. This implies that the operations became more efficient not only from a labor perspective but also when considering the combined cost of labor and equipment.
