Question 1179568
**1. Define the Objective Function**

The objective is to minimize total faculty salaries.  

* Cost of undergraduate courses: RM2500 per course
* Cost of graduate courses: RM3000 per course

Therefore, the objective function is:

**Minimize Z = 2500U + 3000G** where:
* Z = Total faculty salaries
* U = Number of undergraduate courses
* G = Number of graduate courses

**2. Formulate the Constraints**

* **Student Demand:**
    * At least 30 undergraduate courses: U ≥ 30
    * At least 20 graduate courses: G ≥ 20

* **Faculty Contracts:**
    * At least 60 total courses: U + G ≥ 60

* **Non-negativity:**
    * U ≥ 0 
    * G ≥ 0 (You can't have negative courses)

**3. Graphical Solution**

* **Plot the Constraints:**
    1. Treat each inequality as an equation and plot the lines on a graph with U on the x-axis and G on the y-axis.
    2. For U ≥ 30, plot the vertical line U = 30 and shade the region to the right.
    3. For G ≥ 20, plot the horizontal line G = 20 and shade the region above.
    4. For U + G ≥ 60, plot the line U + G = 60 (it intersects the axes at U=60 and G=60) and shade the region above the line.

* **Identify the Feasible Region:** The feasible region is the area where all shaded regions overlap. This represents all the combinations of U and G that satisfy the constraints.

* **Find the Corner Points:** The feasible region will be a polygon.  Identify the coordinates of each corner point of this polygon.

* **Evaluate the Objective Function:** Substitute the U and G values of each corner point into the objective function (Z = 2500U + 3000G).  The corner point that gives the minimum value for Z is the optimal solution.

**4. Interpretation and Report**

Let's assume you've done the graphing and found the optimal solution. Here's how you might structure a report:

**Report: Optimization of Faculty Course Offerings**

**Objective:**

To determine the optimal number of undergraduate (U) and graduate (G) courses to offer in the first semester of 2020/2021, minimizing total faculty salary costs while meeting student demand and faculty contract requirements.

**Methodology:**

Linear programming with a graphical solution approach was used. The objective function and constraints were defined as follows:

* **Objective Function:** Minimize Z = 2500U + 3000G
* **Constraints:**
    * U ≥ 30
    * G ≥ 20
    * U + G ≥ 60
    * U, G ≥ 0

The feasible region was graphically determined, and the objective function was evaluated at each corner point.

**Results:**

*(Here, you would state the optimal solution you found graphically, for example: U = 40, G = 20)*

The optimal solution is to offer 40 undergraduate courses and 20 graduate courses. This will minimize total faculty salaries to RM [insert calculated minimum cost].

**Conclusion:**

By offering 40 undergraduate and 20 graduate courses, the Faculty of Management and Information Technology can meet student demand, fulfill faculty contract obligations, and minimize salary costs.

**Recommendations:**

* The faculty should plan its course offerings accordingly.
* This analysis can be revisited if student demand or faculty contract requirements change.
* Further analysis could incorporate other factors, such as classroom availability and resource allocation.