Question at position 2 P2. Scoreboard Installation You are a project manager working for the Cleveland Indians. Your project consists of installing a new scoreboard at Progressive Fields before Opening Day. You would like to find the minimum-cost schedule for the project. There is a $2,575 per week penalty cost for each week the project is delayed beyond 25 weeks. In addition, your project team determined that indirect costs are $3,500 per week. The minimum cost schedule is _________ weeks (enter response as integer).P2. Scoreboard Installation You are a project manager working for the Cleveland Indians. Your project consists of installing a new scoreboard at Progressive Fields before Opening Day. You would like to find the minimum-cost schedule for the project. There is a $2,575 per week penalty cost for each week the project is delayed beyond 25 weeks. In addition, your project team determined that indirect costs are $3,500 per week. The minimum cost schedule is _________ weeks (enter response as integer).[input]

Question

Question at position 2 P2. Scoreboard Installation You are a project manager working for the Cleveland Indians. Your project consists of installing a new scoreboard at Progressive Fields before Opening Day. You would like to find the minimum-cost schedule for the project. There is a $2,575 per week penalty cost for each week the project is delayed beyond 25 weeks. In addition, your project team determined that indirect costs are $3,500 per week. ​The minimum cost schedule is _________ weeks (enter response as integer).P2. Scoreboard Installation You are a project manager working for the Cleveland Indians. Your project consists of installing a new scoreboard at Progressive Fields before Opening Day. You would like to find the minimum-cost schedule for the project. There is a $2,575 per week penalty cost for each week the project is delayed beyond 25 weeks. In addition, your project team determined that indirect costs are $3,500 per week. ​The minimum cost schedule is _________ weeks (enter response as integer).[input]

BlackTom AI · Accepted Answer

First, restate the problem to focus on the objective: minimize total project cost given a weekly indirect cost and a penalty that kicks in only after week 25.

Let D be the project duration in weeks. The total cost C(D) consists of two parts:
- Indirect costs: 3500 per week for every week of the project, so 3500D.
- Penalty: 2575 per week for each week beyond 25, which is 0 if D ≤ 25, and 2575(D−25) if D > 25.

This yields a piecewise cost function:
- If D ≤ 25: C(D) = 3500D.
- If D > 25: C(D) = 3500D + 2575(D−25) = 6075D − 64375.

Now, compare the slopes of the two regions to understand how cost changes with D.
- For D ≤ 25, the slope is 3500 per additional week.
- For D > 25, the slope is 6075 per additional week, which is steeper than 3500.
Because the cost function has a smaller slope up to week 25 and then a steeper slope after week 25, the minimum occurs at the corner where the two pieces meet, i.e., at D = 25.

Evaluating the two nearby options:
- At D = 25 (no penalty yet): C(25) = 3500 × 25 = 87,500.
- At D = 26 (one week beyond 25): C(26) = 6075 × 26 − 64,375 = 157,950 − 64,375 = 93,575. Alternatively, using the piecewise form: C(26) = 3500×26 + 2575×1 = 91,000 + 2,575 = 93,575.
We can see that C(26) > C(25), and the cost would continue to increase by 6075 for each additional week beyond 25.

Therefore, any schedule longer than 25 weeks increases total cost, and any schedule shorter than 25 weeks would reduce the time but cannot reduce the indirect costs below 25 weeks because the indirect cost accrues every week and there is no savings from shortening within the ≤25 region beyond reducing D in the linear 3500D relationship. The minimum occurs exactly at D = 25 weeks, where there is no penalty and only indirect costs.

Thus, the minimum-cost schedule is 25 weeks. (If you were to enter this as an integer in the blank, you would input 25.)

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