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 members, 45 administrative staff, 30 classrooms, 60 computers, GHc170,000 scholarship funding, and GHc80,000 research funding. Professional development programs: 45 faculty members, 35 administrative staff, 35 classrooms, 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 outlines a linear programming problem faced by the University of Ghana Business School regarding resource allocation and production planning for four product offerings. It provides required data such as unit revenue contributions, available resources, production requirements for each product, and asks specific questions based on a sensitivity report related to optimal solutions, resource constraints, and revenue calculations.
Answer
The specific optimal solution cannot be determined without the numerical data for revenues and constraints.
Answer for screen readers
The final optimal solution will depend on the specific numerical values provided in the problem. Without these values, we cannot provide a specific total revenue or product distribution. If provided, insert them into the objective function and constraints to calculate the numerical answer.
Steps to Solve
-
Identify the Variables
Define the variables for the linear programming problem. For example, let:
- $x_1$: Number of units of Product 1 to produce
- $x_2$: Number of units of Product 2 to produce
- $x_3$: Number of units of Product 3 to produce
- $x_4$: Number of units of Product 4 to produce
-
Formulate the Objective Function
The objective function represents the total revenue generated by all products. Based on the given unit contributions, construct the function:
$$ Z = c_1 x_1 + c_2 x_2 + c_3 x_3 + c_4 x_4 $$
where $c_1$, $c_2$, $c_3$, and $c_4$ are the unit revenue contributions for each product. -
Set Up Constraints
Identify the constraints based on the available resources and production requirements. Each constraint can be expressed as:
- $a_1 x_1 + a_2 x_2 + a_3 x_3 + a_4 x_4 \leq R$
where $a_i$ are the resources needed for each product and $R$ is the available resource constraint.
-
Non-negativity Constraints
Ensure that all variable values are non-negative:
$$ x_1, x_2, x_3, x_4 \geq 0 $$ -
Solve the Linear Program
Use graphical methods or the Simplex method to find the optimal values of $x_1$, $x_2$, $x_3$, and $x_4$ that maximize the objective function $Z$. -
Analyze Sensitivity Report
Review the sensitivity report to interpret how changes in the constraints or coefficients affect the optimal solution and overall revenue.
The final optimal solution will depend on the specific numerical values provided in the problem. Without these values, we cannot provide a specific total revenue or product distribution. If provided, insert them into the objective function and constraints to calculate the numerical answer.
More Information
Linear programming is a valuable technique for maximizing profits or minimizing costs based on given constraints. Sensitivity analysis helps in understanding how sensitive the optimal solution is to changes in parameters.
Tips
- Forgetting to include non-negativity constraints can lead to invalid solutions.
- Misinterpreting constraints often results in incorrect objective function values.
- Skipping steps in solving the linear program can cause confusion and errors in the final answer.
AI-generated content may contain errors. Please verify critical information
