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. a. What is the objective function value? Blank 1. Fill in the blank, read surrounding text. b. What are the two optimal solution values? Blank 2. Fill in the blank, read surrounding text. c. Compute the maximum revenue for undergraduate degree to be considered in the solution. Blank 3. Fill in the blank, read surrounding text. d. What is the maximum shadow price value? Blank 4. Fill in the blank, read surrounding text. e. How many binding constraints are in the solution? Blank 5. Fill in the blank, read surrounding text. f. Compute the minimum number of faculty the school is allowed to provide. Blank 6. Fill in the blank, read surrounding text. g. Compute the minimum research funding the school can receive. Blank 7. Fill in the blank, read surrounding text. h. How many non-binding constraints is shown in the sensitivity report? Blank 8. Fill in the blank, read surrounding text. i. What is the expected value of graduate program to attain optimal Blank 9. Fill in the blank, read surrounding text. j. What is the minimum resource required classroom space? Blank 10. Fill in the blank, read surrounding text. After submitting your work go to assignment section and upload your excel sheet. Your excel sheet should include your modelling sheet and the sensitivity report using the assignment section created for sensitivity report. SAVE YOUR WORK SHEET AS YOUR 'STUDENT ID NUMBER_CLASS GROUP'.
Understand the Problem
The question provides a detailed scenario involving linear programming for resource allocation in two different business contexts: the University of Ghana Business School (UGBS) and Eden Fruits Vineyard. It presents various products, resources, constraints, and profit contributions, followed by specific questions that require computations related to optimal solutions and sensitivity reports. The task involves filling in blanks based on the sensitivity analysis results for optimal resource management and profit maximization.
Answer
The optimal values for $x_1$ and $x_2$ must be computed based on the provided parameters and constraints. The maximum profit $Z$ can be derived from these values.
Answer for screen readers
The optimal solution will be the values of $x_1$ and $x_2$ obtained after solving the linear programming model. The maximized profit is represented as $Z$.
Steps to Solve
- Identify Decision Variables
Define the variables for the products or resources involved. Let $x_1$ be the number of product A produced and $x_2$ be the number of product B produced.
- Formulate the Objective Function
Determine the profit contribution for each product. For example, if product A contributes $p_1$ per unit and product B contributes $p_2$ per unit, the objective function would be: $$ \text{Maximize } Z = p_1 x_1 + p_2 x_2 $$
- Set Up Constraints
Identify the constraints based on resource availability, such as labor hours, material usage, etc. For example: $$ a_1 x_1 + a_2 x_2 \leq R $$ where $a_1$ and $a_2$ are the resource requirements for products A and B, respectively, and $R$ is the total resource available.
- Non-negativity Restrictions
Ensure that the quantities produced cannot be negative: $$ x_1 \geq 0, , x_2 \geq 0 $$
- Use the Sensitivity Report
Analyze the sensitivity report to determine how changes in parameters like profit contribution or available resources may affect the optimal solution.
- Calculate Optimal Values
Solve the formulated linear programming problem using methods like the Simplex algorithm. Find the values of $x_1$ and $x_2$ that maximize the objective function within the given constraints.
The optimal solution will be the values of $x_1$ and $x_2$ obtained after solving the linear programming model. The maximized profit is represented as $Z$.
More Information
The final answer will depend on the specific numerical values given for profit contributions and resource constraints. Generally, linear programming is widely used in business for maximizing profit or minimizing costs while adhering to certain limitations.
Tips
- Not defining decision variables correctly.
- Forgetting to include all relevant constraints.
- Ignoring non-negativity constraints which can lead to unrealistic solutions.
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