Cody Parker is a senior manager who is evaluating four possible projects (A, B, C, and D). Cody can fund any project in part or in total; however, he cannot change the funding percent once the project has started. The costs and revenues associated with any project are a linear function of the funding percent; for example, if Project A is funded at 40 percent, it would cost (.4 x $14 =) $5.6 in year 1 and generate a value to the company's portfolio of (.4 x 10 = ) 4. Also, once a project has started, it cannot be stopped until completed.

If the projects are fully funded (at 100 percent), the cash flows for each project over the next three years are indicated in the table below. The positive numbers in the table indicate expenses (cash flows out) while the negative numbers indicate payments to the company (cash flows in). In addition, each project has been valued using a ranking/scoring method described in this chapter; the maximum value of each project is indicated in the table (e.g., Project A would be worth 10 points if fully funded).

Project A is a utility project to repair and upgrade the maintenance facility but it will not generate any positive cash flow to the company (thus, only costs are associated with that project). The other three projects represent new products or services that will generate positive cash flows to the company in the third year.

Cody wants to maximize the value of his company's project portfolio over the next three years since that will determine his bonus. However, Cody has stated that the total cash flows in each year cannot exceed the budget constraint that was specified by the company's CFO. For example, no more than $50 can be spent in Year 1 as indicated by the budget constraint for Year 1.

Cody wants to select a project portfolio that maximizes the total portfolio score, subject to the budget constraints. Each project can be funded between zero percent (no funding) to a maximum of 100 percent (fully funded).

a) Given the budget constraints, which projects should Cody recommend for funding? At what level?

b) Cody's boss has reconsidered Project A, the utility project to repair the roof. She now feels that only offers a value of 2.0 if fully funded (not 10.0). How does this change the composition of the project portfolio?

You have been given a small project to manage that your boss feels is important. Six tasks have been identified that define the project; the expected duration and precedence relationships between these tasks are indicated in the AON network below.

**a) **Assuming that all precedence relationships are Finish-to-Start, use the Critical Path Method (CPM) (e.g., forward pass, backward pass) to find the early start times, early finish times, late start times, late finish times, and total slacks (floats) for each task. Which tasks are critical? (Use the template provided for this problem.)

**b) **Formulate the "Mark" linear programming model using Solver for finding the critical path. Use the template provided for this problem; click on the tab marked "Mark Model". You will have to define the constraints in the Solver dialog box as well as define the objective function cell. Use this model to find the earliest starting times for the tasks in this project AND the latest starting times for the tasks in this project (you will have to modify the objective function appropriately). Using these early start and late start times, compute the early f and late finish times and total slack values for each task. Indicate which tasks lie on the critical path. Compare your work to the values you found in part (a).

**c) **Formulate the "Gary" linear programming using Solver model to replicate the calcuations in part (b). Use the template provided for this problem; click on the tab marked "Gary Model".

**d) **Analyze this project using MS Project 2013 or 2016. Show that the critical path, slack values, etc. correspond to the values calculated in parts (a) and (b) above. Make sure that your MS Project output shows the total slack values as well as the start and finish dates, etc. Using the early start dates, develop a Gantt chart for this project. Take a screen shot of your input and Gantt chart and paste this into your solution.

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