To solve this problem step by step:

Part (a)

Step 1: Calculate the expected net present value (NPV) for each alternative.

Given probabilities for the demand scenarios are:

• Demand rising faster: $60\%$ or $0.6$
• Demand rising at the current rate: $30\%$ or $0.3$
• Demand increasing at a slower rate or falling: $10\%$ or $0.1$

We calculate the expected NPV for each alternative:

1. Expand: $\text{Expected NPV} = (500 \times 0.6) + (400 \times 0.3) + (-150 \times 0.1)$ $= 300 + 120 - 15 = 405$

2. Build a new plant: $\text{Expected NPV} = (700 \times 0.6) + (200 \times 0.3) + (-300 \times 0.1)$ $= 420 + 60 - 30 = 450$

3. Subcontract: $\text{Expected NPV} = (200 \times 0.6) + (150 \times 0.3) + (-50 \times 0.1)$ $= 120 + 45 - 5 = 160$

Step 2: Determine the best course of action.

The highest expected NPV is $450$ for "Build a new plant." Therefore, the company should choose to build a new plant.

Part (b)

Step 1: Use the forecast probabilities to adjust the decision.

The forecast suggests demand will rise at the current rate. The revised probabilities (from the forecast) are:

• Probability of actual demand rising faster: $P = 0.3$
• Probability of actual demand continuing at the current rate: $P = 0.7$
• Probability of actual demand increasing slower/falling: $P = 0.4$

Now, we re-calculate the expected NPV based on these probabilities.

1. Expand: $\text{Expected NPV} = (500 \times 0.3) + (400 \times 0.7) + (-150 \times 0.4)$ $= 150 + 280 - 60 = 370$

2. Build a new plant: $\text{Expected NPV} = (700 \times 0.3) + (200 \times 0.7) + (-300 \times 0.4)$ $= 210 + 140 - 120 = 230$

3. Subcontract: $\text{Expected NPV} = (200 \times 0.3) + (150 \times 0.7) + (-50 \times 0.4)$ $= 60 + 105 - 20 = 145$

Step 2: Determine the new best course of action.

The highest expected NPV is now $370$ for "Expand the existing plant." Based on the revised probabilities, the company should choose to expand the existing plant.

Summary of Decisions:

1. Part (a): Build a new plant (Expected NPV: 450).
2. Part (b): Expand the existing plant (Revised Expected NPV: 370).

Let me know if you'd like detailed explanations for any part of the calculations.

Questions to Explore Further:

1. How would changes in the probabilities impact the expected NPV for each alternative?
2. What are the risks of overestimating or underestimating demand scenarios in this context?
3. How might additional forecast data (e.g., economic trends) refine the decision-making process?
4. Would including a risk premium or discount rate alter the recommended decision?
5. How sensitive is the optimal choice to variations in the estimated NPVs?

Tip:

Always validate decision-making models with real-world data and sensitivity analysis to account for uncertainty and changing market conditions.