Linear programming is a powerful tool that can be used by university administrations to make optimal decisions regarding resource allocation and production planning. In this proble... Linear programming is a powerful tool that can be used by university administrations to make optimal decisions regarding resource allocation and production planning. In this problem, we consider a linear programming problem faced by the University of Ghana Business School (UGBS) administration. The UGBS offers four main products: undergraduate degrees (x1), graduate degrees (x2), online courses (x3), and professional development programs (x4). Each product has specific requirements in terms of resources, such as faculty, staff, classroom space, technology, and funding. To maximize revenue while managing resources effectively, the UGBS must find the right balance of product offerings. Unit Revenue contributions per each product offering are 750, 1200, 800 and 11000 for undergraduate degrees, graduate degrees, online courses, and professional development programs, respectively. The UGBS has six resource constraints, namely; faculty, staff, classroom space, technology, funding for scholarships and funding for research. It has the following amount of each of these resources available. Faculty: 250 professors, Staff: 75 administrative staff, Classroom space: 90 classrooms, Technology: 185 computers, Funding for scholarships: GHc500,000, Funding for research: GHc350,000 The UGBS’s production requirements for each product are: Undergraduate degrees: 90 faculty members, 30 administrative staff, 25 classrooms, 65 computers, GHc120,000 scholarship funding, and GHc95,000 research funding. Graduate degrees: 80 faculty members, 25 administrative staff, 20 classrooms, 65 computers, GHc180,000 scholarship funding, and GHc120,000 research funding. Online courses: 100 faculty member, 45 administrative staff, 30 classroom, 60 computers, GHc170,000 scholarship funding, and GHc80,000 research funding. Professional development programs: 45 faculty members, 35 administrative staff, 35 classroom, 50 computers, GHc50,000 scholarship funding and GHc75,000 research funding. USE YOUR SENSITIVITY REPORT TO ANSWER ALL THE QUESTIONS BELOW. USE 2 DECIMAL PLACES FOR ALL COMPUTATIONS AND ANSWERS TO 2 DECIMAL PLACE WHERE REQUIRED.
Understand the Problem
The question presents a linear programming scenario focusing on resource allocation for the University of Ghana Business School and Eden Fruits Vineyard. It outlines specific product offerings, resource constraints, and profit margins. The questions that follow require the application of sensitivity analysis to determine optimal solutions, profit contributions, and constraint characteristics.
Answer
The optimal solution will be determined based on specific values in the problem statement, but it involves maximizing $Z = p_A x_1 + p_B x_2$ subject to given constraints.
Answer for screen readers
The final answer will depend on the specific values of profit margins and resources provided in the problem statement.
Steps to Solve
-
Define the Decision Variables
Let $x_1$ represent the number of product A units allocated to the University of Ghana Business School and $x_2$ represent the number of product B units allocated to Eden Fruits Vineyard. -
Set Up Objective Function
Formulate the objective function to maximize profit. If the profit per unit for product A is $p_A$ and for product B it is $p_B$, then the function can be expressed as:
$$ Z = p_A x_1 + p_B x_2 $$ -
Establish Constraints
Identify the constraints based on resource limits. For example, if resource 1 has a limit $R_1$ pertaining to $x_1$ and $x_2$, this can be expressed as:
$$ a_1 x_1 + b_1 x_2 \leq R_1 $$
Repeat for other resources accordingly. -
Graph the Constraints
If needed, graph the constraints on a coordinate plane to visualize the feasible region. This helps in understanding the areas where solutions may exist. -
Identify Corner Points
Determine the corner points of the feasible region. These points arise from the intersections of the constraints, and they are the candidates for optimal solutions. -
Evaluate the Objective Function at Corner Points
Substitute the corner points into the objective function to find the profit at each point. For example, for each corner point $(x_1, x_2)$, calculate:
$$ Z = p_A x_1 + p_B x_2 $$ -
Select the Optimal Solution
Identify which of the calculated profits is the highest. The corresponding corner point is the optimal allocation of resources. -
Perform Sensitivity Analysis
Analyze how changes in the constraints affect the optimal solution. Adjust the parameters and observe the impact on the profit and allocations.
The final answer will depend on the specific values of profit margins and resources provided in the problem statement.
More Information
In linear programming, the optimal solution can change when there are shifts in the parameters of the objective function or constraints. Sensitivity analysis helps to understand how stable the solution is under such changes, and it can guide decision-makers regarding potential resource adjustments.
Tips
- Misinterpreting constraints can lead to incorrect feasible regions. Always verify that all constraints are correctly defined and plotted.
- Failing to check all corner points may result in overlooking the point with maximum profit. Ensure to evaluate every corner point within the feasible region.
- Overlooking sensitivity analysis by assuming the optimal solution is unchanged with parameter variations. Recognizing which parameters impact the solution is crucial.
AI-generated content may contain errors. Please verify critical information
