(a) Consider the following multi-period capital budgeting problem. An investor is consid- ering 8 independent investment...

     

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(a) Consider the following multi-period capital budgeting problem. An investor is consid- ering 8 independent investment projects. Each project requires full participation, no partial participation is allowed. If selected, a project may require cash contributions yearly over the next 3 years, as in the following table. There is an external fixed amount of $1,000,000 available for these investments in each of the following 3 years. Each investment project is expected to have a net return as shown in Table 1. The investor needs to determine the sub- set of projects to invest in, so as to maximize the total expected net returns. For simplicity, assume that: • Unutilized funds are not carried forward to the next period and are not available for investments. • Cash flows are not discounted for the time value of money (we could consider discount- ing of cash flows, but the problem becomes considerably more difficult to formulate). cash out-flows P₁ P₂ P3 P₁ P5 P6 P7 P8 20 40 50 25 15 7 23 13 30 20 30 25 25 22 23 28 10 0 10 35 30 23 23 15 Expected Net Return 10 15 20 20 15 13 13 14 Table 1: All figures in $10,000's of dollars Year 1 Year 2 Year 3 Funds Available 100 100 100 Formulate the problem as a binary programming problem, and use either Python or Excel to solve it. Hint: Introduce 8 binary variables and consider 3 constraints (knapsacks). (b) Consider the same setting as in Problem (a), but relax the first assumption, i.e., unused funds can be carried forward to the next period and are available for cash out-flows. Formulate the problem as an integer programming problem, and use either Python or Excel to solve it. Compare your solution to the one from Problem (a). (a) Consider the following multi-period capital budgeting problem. An investor is consid- ering 8 independent investment projects. Each project requires full participation, no partial participation is allowed. If selected, a project may require cash contributions yearly over the next 3 years, as in the following table. There is an external fixed amount of $1,000,000 available for these investments in each of the following 3 years. Each investment project is expected to have a net return as shown in Table 1. The investor needs to determine the sub- set of projects to invest in, so as to maximize the total expected net returns. For simplicity, assume that: • Unutilized funds are not carried forward to the next period and are not available for investments. • Cash flows are not discounted for the time value of money (we could consider discount- ing of cash flows, but the problem becomes considerably more difficult to formulate). cash out-flows P₁ P₂ P3 P₁ P5 P6 P7 P8 20 40 50 25 15 7 23 13 30 20 30 25 25 22 23 28 10 0 10 35 30 23 23 15 Expected Net Return 10 15 20 20 15 13 13 14 Table 1: All figures in $10,000's of dollars Year 1 Year 2 Year 3 Funds Available 100 100 100 Formulate the problem as a binary programming problem, and use either Python or Excel to solve it. Hint: Introduce 8 binary variables and consider 3 constraints (knapsacks). (b) Consider the same setting as in Problem (a), but relax the first assumption, i.e., unused funds can be carried forward to the next period and are available for cash out-flows. Formulate the problem as an integer programming problem, and use either Python or Excel to solve it. Compare your solution to the one from Problem (a).

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a Formulating the problem as a binary programming problem Lets define the decision varia... View the full answer