1. **Problem Statement:** Calculate average labour hours per unit for 1, 2, 4, and 8 units, total cumulative labour hours for 8 units, and total labour cost using an 80% learning curve with initial unit time 600 hours. Also explain factors causing deviations and managerial applications. 2. **Learning Curve Model:** The average time per unit when cumulative output doubles is multiplied by the learning rate (80% or 0.8). 3. **Calculate average labour hours per unit:** The formula for average time per unit at cumulative output $N$ is: $$T_N = T_1 \times N^{\log_2 b}$$ where $T_1=600$ hours, $b=0.8$ (learning rate). Calculate $\log_2 0.8$: $$\log_2 0.8 = \frac{\ln 0.8}{\ln 2} \approx \frac{-0.2231}{0.6931} \approx -0.3219$$ Now calculate average times: - For 1 unit: $$T_1 = 600 \times 1^{-0.3219} = 600$$ - For 2 units: $$T_2 = 600 \times 2^{-0.3219} = 600 \times 0.8 = 480$$ - For 4 units: $$T_4 = 600 \times 4^{-0.3219} = 600 \times 0.8^2 = 600 \times 0.64 = 384$$ - For 8 units: $$T_8 = 600 \times 8^{-0.3219} = 600 \times 0.8^3 = 600 \times 0.512 = 307.2$$ 4. **Calculate total cumulative labour hours for 8 units:** Total time is sum of times for each unit. Using the cumulative average time formula: $$Y_N = T_1 \times \frac{N^{\log_2 b + 1}}{N} = T_1 \times N^{\log_2 b}$$ But more straightforward is to use the cumulative average time formula: $$Y_N = T_1 \times N^{\log_2 b}$$ which is the average time per unit for $N$ units. Total cumulative hours for $N$ units: $$C_N = Y_N \times N = T_1 \times N^{\log_2 b} \times N = T_1 \times N^{1 + \log_2 b}$$ Calculate exponent: $$1 + \log_2 0.8 = 1 - 0.3219 = 0.6781$$ Calculate total hours for 8 units: $$C_8 = 600 \times 8^{0.6781}$$ Calculate $8^{0.6781}$: $$8^{0.6781} = e^{0.6781 \times \ln 8} = e^{0.6781 \times 2.0794} = e^{1.409} \approx 4.09$$ So: $$C_8 = 600 \times 4.09 = 2454$$ hours (approx) 5. **Calculate total labour cost:** Labour cost per hour = 500 Total cost: $$\text{Cost} = 2454 \times 500 = 1,227,000$$ 6. **Factors causing deviations from theoretical learning curve:** - Variability in worker skill levels and training. - Changes in production methods or technology. - Interruptions or delays in supply chain or equipment. 7. **Managerial applications of learning curve:** - Cost estimation: Predict future labour costs as production increases. - Budgeting: Plan budgets based on expected reductions in labour hours. - Production planning: Schedule resources and timelines considering improved efficiency.