A certain academic department at a local university will conduct a research project. The department will need to hire graduate research assistants and professional researchers. Each graduate research assistant will need to work 23 hours per week on fieldwork and 17 hours per week at the university's research center. Each professional researcher will need to work 15 hours per week on fieldwork and 25 hours per week at the university's research center. The minimum number of hours needed per week for fieldwork is 154 and the minimum number of hours needed per week at the research center is 140. Each research assistant will be paid $295 per week and each professional researcher will be paid $416 per week. Let x denote the number of graduate research assistants hired and let y denote the number of professional researcher hired. The department wants to minimize cost. Set up the Linear Programming Problem for this situation.

First, to find the Objective function of the linear programming problem we need to note that we are told, "the department wants to minimize cost (weekly cost)," meaning our objective function is Min C (Cost) = 295x+ 416y. Where x≥ 0, y≥ 0.

Constraints:

Let x denote the number of graduate research assistants hired and let y denote the number of professional researchers hired.

Note these statements that form the constraints for the different types of researchers:

1. Fieldwork hours constraints:

Each graduate research assistant will need to work 23 hours per week.
Each professional researcher will need to work 15 hours per week on fieldwork.

And the minimum (≥) number of hours needed per week for fieldwork is 154. Which implies this model, 23x+ 15y≥ 154.

2. University's research center hours constraints:

Each graduate research assistant will need to work 17 hours per week at the university's research center.
Each professional researcher will need to work 25 hours per week at the university's research center.

And the minimum (≥) number of hours needed per week at the research center is 140. Which implies this model, 17x+ 25y≥ 140.