The Dean Of The Western College Of Business Must Plan The Sc

The Dean Of The Western College Of Business Mustplan The Schools

The dean of the Western College of Business must plan the school's course offerings for the fall semester. 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 costs the college an average of $2,500 in faculty wages, and each graduate course costs $3,000. How many undergraduate and graduate courses should be taught in the fall so that total faculty salaries are kept to a minimum? Minimize Z = 160000 by changing, X = 40, Y = 20 subject to, 60 ≥ 20.

Paper For Above instruction

The planning and optimization of course offerings in higher education institutions involve complex decision-making processes that aim to balance student demand, faculty constraints, and financial considerations. The case of the Western College of Business exemplifies this challenge, requiring strategic planning to minimize costs while satisfying student and contractual requirements. This paper examines the linear programming problem faced by the college, analyzes its underlying assumptions, and explores solutions based on optimization techniques.

Introduction

In higher education administration, resource allocation must be both efficient and effective. The Western College of Business faces a specific scenario where it must determine the optimal number of undergraduate and graduate courses to offer. The key objectives are to meet student demand, adhere to faculty contracts, and minimize faculty wages. The problem is modeled using linear programming, a mathematical approach suitable for such resource allocation challenges.

Problem Description

The institution's constraints include offering at least 30 undergraduate courses and 20 graduate courses. The total courses offered must be at least 60, considering contractual obligations and student demand. Each undergraduate course costs $2,500, whereas each graduate course costs $3,000 in faculty wages. The goal is to find the number of courses, denoted by variables X (undergraduate courses) and Y (graduate courses), that minimize the total wages, expressed as:

\[ Z = 2,500X + 3,000Y \]

subject to the constraints:

\[ X \geq 30 \]

\[ Y \geq 20 \]

\[ X + Y \geq 60 \]

Additionally, the problem states a solution with initial values X=40 and Y=20, yielding a total cost of $160,000, which suggests that the problem is to verify or identify the optimal solution within the constraints.

Methodology

Applying linear programming, the solution involves graphing the constraints and identifying the feasible region where all conditions overlap. The optimal point on this feasible region minimizes the total faculty wages. The corner points of the feasible region are examined, and the minimum cost is determined at one of these points using the objective function.

Given the constraints, the corner points are:

1. (30,30): satisfies all constraints

2. (40,20): as given

3. (30,30): same as 1, already satisfying the minimum course requirements

4. Intersection point where \( X + Y=60 \):

- If \( X=30 \), then \( Y=30 \)

- If \( Y=20 \), then \( X=40 \)

Calculating the total wages at these points:

- At (30,30): \( Z = 2,500 \times 30 + 3,000 \times 30 = 75,000 + 90,000 = 165,000 \)

- At (40,20): \( Z = 2,500 \times 40 + 3,000 \times 20 = 100,000 + 60,000 = 160,000 \)

Between these options, the minimum is at (40,20), with a total cost of $160,000. Thus, the optimal solution aligns with the current proposed plan, suggesting that offering 40 undergraduate and 20 graduate courses minimizes total faculty salaries within the constraints.

Analysis & Implications

This result indicates that increasing undergraduate courses to 40 and keeping graduate courses at 20 is optimal under the given costs and constraints. It also emphasizes the importance of linear programming in decision-making, allowing administrators to make data-driven choices that optimize financial outcomes while satisfying operational constraints.

In addition, real-world decision-making might incorporate further considerations such as faculty workload, course capacity, and departmental priorities, which could complicate the model. Nevertheless, the LP approach provides a clear starting point for such strategic planning.

Conclusion

Effective resource allocation in academic institutions can be achieved through linear programming models that consider demand constraints, contractual obligations, and cost minimization objectives. For the Western College of Business, the optimal strategy involves offering 40 undergraduate and 20 graduate courses, aligning with the minimal faculty salary expenditure of $160,000. Incorporating such quantitative methods into planning processes ensures efficient use of resources, supports institutional goals, and enhances decision-making efficacy.

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