Optimize profitability for coding contract decisions

A client is offering two fixed-price projects, mutually exclusive, that begin on 2025-10-01 and must be completed by 2025-10-31. Your team currently consists of 3 engineers.

Costs and productivity:

• Base pay is $70 per hour per engineer, with overhead equal to 25% of labor cost.
• Regular time allows up to 40 hours per engineer per week, at a rate of 25 lines per hour.
• Overtime allows up to 20 additional hours per engineer per week, paid at 1.5 times the base rate; overtime output is 20 lines per hour.
• An optional contractor costs $110 per hour, produces 22 lines per hour, requires 10 hours of onboarding at half productivity, cannot work overtime, and carries a 10% agency fee on labor cost.

Quality: 8% of all produced lines require rework, which is done at the same productivity and is not paid by the client; rework is discovered within the month.

Client options:

• Project A: 1,000 lines; pays $45 per line; on-time bonus $5,000; late penalty $3,000 per week.
• Project B: 2,000 lines; pays $40 per line; on-time bonus $12,000; the same late penalty applies.

Assume working days are distributed evenly across the month and that rework must also finish by 2025-10-31 for the bonus to apply.

a) With only the current staff (no contractor), calculate completion time and total profit for Projects A and B, both with and without overtime; specify any assumptions about how overtime hours are allocated.

b) If one contractor can be added beginning 2025-10-07, recalculate completion time and profit for A and B, including onboarding time and the agency fee.

c) Which project choice maximizes expected profit while still meeting the deadline with at least a 95% buffer against a ±10% variation in productivity?

d) Find the break-even payment per line for Project B at which B becomes more attractive than A under the best staffing arrangement.

e) Sensitivity: describe how profit changes as the rework rate varies from 0% to 15%. Show the formulas and provide final numeric recommendations.

Overview: The question tests skills in quantitative cost modeling, scheduling, and probabilistic sensitivity analysis, within statistics and math, with emphasis on profit optimization and resource-allocation decisions relevant to a data scientist position.