Optimization Tasks

Optimization Tasks#

Choose one task for your group. Your goal is not to solve the problem numerically, but to formulate it as an optimization problem that could be solved in principle.

For your chosen task you need to identify:

  • the decision variables

  • the objective function

  • the main constraints

  • whether the problem appears linear, quadratic, convex, nonconvex, or unclear

  • what mathematical or computational tools you would try first

Find a group of 3-4 people and sign up for an unclaimed option on the slides.

By the end of class, include information about your question and the bullet points above to the slide deck.

1. Hospital Staff Scheduling#

A hospital needs to assign nurses, residents, or other staff members to shifts over the next week. The schedule must cover all required shifts while respecting limits on total hours, required certifications, and fairness across employees.

Questions to get you started:

  • What variables represent the schedule?

  • Is fairness part of the objective or part of the constraints?

  • Does this model stay continuous, or does it naturally become integer-valued?

2. Warehouse Placement for Regional Deliveries#

A company wants to place a warehouse or fulfillment center so that it can serve several cities or stores efficiently. The goal is to reduce delivery costs or average distance while still meeting service requirements.

Questions to get you started:

  • What are the variables if the location is allowed to vary continuously?

  • How would you model travel cost or distance?

  • What assumptions make the model easier to analyze?

3. Advertising Budget Allocation#

A small company has a fixed advertising budget and wants to split it across platforms such as search ads, social media, streaming, and print. The goal is to maximize engagement, conversions, or reach.

Questions to get you started:

  • What are the decision variables?

  • What constraints come from the total budget or minimum platform usage?

  • When would this model be linear, and when might it become nonlinear?

4. Portfolio Allocation with Risk Control#

An investor wants to divide their money among several assets. They want a balance between expected return and risk, and they may also want limits on how much can be invested in any one asset.

Questions to get you started:

  • What variables represent the allocation?

  • How would you model risk?

  • Does the resulting objective look linear, quadratic, or something else?

5. Electric Vehicle Charging Schedule#

A company manages a small fleet of electric vehicles and needs to decide when each vehicle should charge overnight. Electricity prices vary over time, and the building has a cap on total power usage.

Questions to get you started:

  • What are the variables over time?

  • What constraints ensure each vehicle has enough charge by morning?

  • Which modeling assumptions keep the problem convex?

6. Crop Planning for a Farm#

A farm has limited land, water, labor, and fertilizer, and must decide how much area to devote to each crop. The goal might be to maximize profit, stabilize yield, or reduce water use.

Questions to get you started:

  • What are the decision variables?

  • Which resources create the main constraints?

  • Would uncertain weather change the way you formulate the model?

7. Student Course Planning#

A student wants to choose courses for the next semester to make progress toward graduation while balancing workload, time conflicts, and personal interests.

Questions to get you started:

  • What variables represent taking or not taking a course?

  • How would you encode prerequisites and time conflicts?

  • Is this the kind of problem that fits neatly into convex optimization?

8. Manufacturing Production Plan#

A factory produces several products and must decide how much of each to make next month. The decision is limited by worker hours, raw materials, machine capacity, and storage.

Questions to get you started:

  • What objective makes sense: profit, cost, or waste reduction?

  • Which constraints are most important?

  • Does this look like a standard linear programming problem?

9. Diet or Meal Planning#

A school, hospital, or athlete wants to design a meal plan that meets nutritional targets at minimum cost. The plan may also include upper and lower bounds on calories, protein, sodium, or other nutrients.

Questions to get you started:

  • What variables represent the food choices?

  • How do nutritional requirements become constraints?

  • What makes this a good candidate for linear optimization?

10. Fitting a Predictive Model to Data#

A data scientist wants to fit a model to data while avoiding overfitting. The goal is to choose parameters that fit observed data well while also penalizing overly large coefficients.

Questions to get you started:

  • What are the optimization variables?

  • How would you write a least-squares or regularized least-squares objective?

  • Why is this especially connected to material from this course?

11. Water Reservoir Management#

A city or utility must decide how much water to release from a reservoir over time. The decision should balance water supply, flood prevention, environmental requirements, and possibly hydropower production.

Questions to get you started:

  • What variables change over time?

  • What conservation or storage constraints are needed?

  • Which parts of the model might be hard to keep convex?

12. Emergency Resource Placement#

A city wants to decide where to place ambulances, emergency supplies, or first-aid stations so that as many people as possible can be reached quickly. The system may also need to account for limited staffing or equipment.

Questions to get you started:

  • What does it mean to optimize coverage or response time?

  • What are the decision variables?

  • Which parts of the problem are geometric, and which parts may be discrete?

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