SOLUTION: Hi! can please help me with this question ? Thankyou! The dean of Faculty of Management and Information Technology must plan the faculty’s course offerings for the first seme

Question 1179568: Hi! can please help me with this question ? Thankyou!
The dean of Faculty of Management and Information Technology must plan the
faculty’s course offerings for the first semester, 2020/2021. Student demands make it necessary to offer at least 30 undergraduate and 20 graduate courses in the term.Faculty contracts also dictate that at least 60 courses be offered in total. Each undergraduate course taught course the faculty an average of RM2,500 in faculty wages, and each graduate course costsRM3,000. By using the graphical method, calculate the number of undergraduate and graduate courses should be taught in the first semester so that the total faculty salaries are kept to a minimum. We let
U = undergraduate courses, and
G = graduate courses.
Note: You might have to prepare a report for proposing an optimal solution by usingquantitative analysis techniques.

Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1987)   (Show Source): You can put this solution on YOUR website!
**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.
Answer by ikleyn(52834)   (Show Source): You can put this solution on YOUR website!
.
Hi! can please help me with this question ? Thankyou!
The dean of Faculty of Management and Information Technology must plan the faculty’s course
offerings for the first semester, 2020/2021. Student demands make it necessary to offer
at least 30 undergraduate and 20 graduate courses in the term. Faculty contracts
also dictate that at least 60 courses be offered in total. Each undergraduate course
taught course the faculty an average of RM2,500 in faculty wages, and each graduate course
costs RM3,000. By using the graphical method, calculate the number of undergraduate
and graduate courses should be taught in the first semester so that the total
faculty salaries are kept to a minimum. We let
U = undergraduate courses, and
G = graduate courses.
Note: You might have to prepare a report for proposing an optimal solution by using quantitative
analysis techniques.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        Surely, this problem can be solved using Linear Programming method.
        But it also can be solved, using logical reasoning and common sense,
        practically mentally, in your mind.

First, notice that if the conditions "at least 30 U and at least 20 G" are satisfied and we have MORE than 60 courses, than the total number of courses can be diminished by 1 such way that we still will be in the feasible domain. Indeed, if U >= 30 and G >= 20, and U + G > 60, it means that at least one number U or G is greater than their corresponding lower boundaries, so this course can be eliminated. Repeating this reasoning as many times as required, we will get the condition U >= 30, G >= 20, U + G = 60 so the total number of courses is precisely 60. Eliminating excessive courses, we diminish the total cost, so it is good, from the problem's point of view. Now, if the number of graduate courses is G, then the number of undergraduate courses is U = 60-G. The total cost is then 2500*U + 3000*G = 2500*(60 - G) + 3000*G = 150000 + 500*G, and we want to make it as small as it is possible under restrictions. From the last formula, it is clear that the total cost is minimum when the number of graduate courses G is at its lower bound G = 20. Thus the optimal solution is G = 20 graduate courses and U = 60-20 = 40 undergraduate courses, giving the minimum possible cost 40*2500 + 20*3000 = 100000 + 60000 = RM160,000.

Solved.

So, if the dean of Faculty of Management and Information Technology has a knack for logic,
he can solve this problem MENTALLY, without using the LP-method.

The idea is to assign as few of expensive courses as possible under restrictions,
and then to add as many of cheaper courses to get other restrictions.

It is another view to the problem, which is always good to have, if possible.

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