The manager's goal is to maximize the Net Present Value (NPV) of the projects while staying within the budget constraint of $1,000,000. Let's analyze the projects with their costs and NPVs, and find which combination maximizes the total NPV without exceeding $1,000,000.

Here are the projects and their details:

Project	Initial Cost	NPV
A	$500,000	$300,000
B	$1,000,000	$400,000
C	$250,000	$200,000
D	$750,000	$550,000
E	$500,000	$250,000

The total budget is $1,000,000.

Step 1: Sort projects by NPV-to-cost ratio

To maximize NPV, it's beneficial to consider the NPV-to-cost ratio:

• Project A: $\frac{300,000}{500,000} = 0.6$
• Project B: $\frac{400,000}{1,000,000} = 0.4$
• Project C: $\frac{200,000}{250,000} = 0.8$
• Project D: $\frac{550,000}{750,000} \approx 0.73$
• Project E: $\frac{250,000}{500,000} = 0.5$

Thus, the best strategy is to select projects that provide the highest NPV-to-cost ratio first, while staying within the budget.

Step 2: Select projects that maximize the total NPV within the budget

• Start with Project C (Cost: $250,000, NPV: $200,000) since it has the highest NPV-to-cost ratio.
• Next, select Project D (Cost: $750,000, NPV: $550,000), as this will maximize the total NPV.

The total cost is $250,000 + $750,000 = $1,000,000, which is within the budget.

Total NPV:

The total NPV for selecting Project C and Project D is: $200,000 + 550,000 = 750,000$

Conclusion:

The manager should select Project C and Project D, giving a total NPV of $750,000.

Would you like further details on how this was calculated or have any other questions?

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Here are 5 questions that build on this information:

1. How does changing the budget constraint affect the project selection?
2. What if the NPV for Project A increases to $400,000? Would this change the decision?
3. What impact does the cost of capital have on the NPV in project selection?
4. Can this problem be solved using a linear programming approach?
5. How does risk factor into the decision when comparing NPVs?

Tip: Always look at NPV-to-cost ratios when trying to maximize returns within a constrained budget!