The human resources department of a medium-sized tech firm has a special team dedicated to creating and implementing learning and development initiatives. The director of HR is particularly concerned with scheduling occurrences of two courses that are mandatory for all new employees: a five-day software intensive in which new employees are given a rigorous introduction to the company’s proprietary software platform, and a two-day organizational culture workshop, in which employees are taught about the company’s history, mission, and values. Both courses are offered semi-regularly to accommodate the inflow of new employees. The director has mandated that over the next six months, the learning and development team must offer at least 8 software intensives and at least 10 culture workshops. Further, the team must offer at least 25 courses total during that time. Both courses are taught by a pool of instructors, which consists of a select group of senior-level employees who take time out of their own work schedules to teach. This pool of instructors has a combined 84 days’ worth of teaching time to offer over the next six months. Although there are multiple instructors in the pool, consider it a single resource: assume that any instructor in the pool can teach any course, and there are no constraints on how courses are allocated across instructors. Each software-intensive costs $5,000 and each culture workshop costs $1,000 in miscellaneous fees, including an offsite facility, catering, and printed materials.

a. Formulate the problem as a linear programming model with the goal of minimizing the total cost the department will incur for offering these courses over the next six months

b. Graph the optimization model, placing the number of software intensives on the horizontal axis, and the number of culture workshops on the vertical axis. Plot and label each constraint and shade the feasible region.

c. Solve for the optimal number of courses the department should offer in order to minimize total cost. Report these values, along with the associated cost for this solution.