Assignment 1: Linear Programming Case Study Your Instructor
Assignment 1 Linear Programming Case Studyyour Instructor Will Assign
Your instructor will assign a linear programming project for this assignment according to the following specifications. It will be a problem with at least three (3) constraints and at least two (2) decision variables. The problem will be bounded and feasible. It will also have a single optimum solution (in other words, it won’t have alternate optimal solutions).
The problem will also include a component that involves sensitivity analysis and the use of the shadow price. You will be turning in two (2) deliverables, a short writeup of the project and the spreadsheet showing your work.
Paper For Above instruction
Linear programming (LP) is a mathematical technique used to optimize a particular objective, subject to certain constraints. This method is fundamental in operations research for decision-making processes involving resource allocation, production scheduling, transportation, and other logistical problems. In this case study, the LP problem is designed to maximize or minimize a specific goal, such as profit or cost, constrained by limited resources.
The problem involves at least three constraints, which could represent resource limitations such as labor hours, raw materials, or budget caps, and has at least two decision variables, which could be quantities of products to produce, staff hours allocated, or other measurable factors. The model is bounded and feasible, meaning solutions exist within the defined constraints, and a unique optimal solution exists, avoiding multiple solutions with equivalent value.
For example, in a manufacturing setting, the decision variables could be the number of units of two products to produce. The objective might be to maximize profit, with constraints representing resource limits—for instance, raw material availability, labor hours, and machine time. The problem's setup entails defining the objective function, such as maximizing total profit, and establishing constraints based on resource consumption, each formulated as linear inequalities. Non-negativity constraints are also included to ensure decision variables are practical (e.g., production quantities cannot be negative).
The LP model could be expressed as follows:
- Objective Function: Maximize Profit = c₁x₁ + c₂x₂
- Subject to Constraints:
- a₁₁x₁ + a₁₂x₂ ≤ b₁
- a₂₁x₁ + a₂₂x₂ ≤ b₂
- a₃₁x₁ + a₃₂x₂ ≤ b₃
- Non-negativity:
- x₁ ≥ 0
- x₂ ≥ 0
After setting up the model, you will employ Excel’s Solver function to find the optimal values of x₁ and x₂ that maximize or minimize the objective. Once the optimal solution is obtained, it’s crucial to interpret what the results indicate in practical terms—such as how many units of each product should be produced to achieve maximum profit under current resource limits.
Sensitivity analysis is an integral part of LP, offering insights into how changes in resource availability or costs affect the optimal solution. The shadow price, derived from Solver, indicates the value of an additional unit of a resource—helping decision-makers understand the worth of relaxing a constraint. Analyzing these values guides strategic decisions, such as investing in additional resources or adjusting operational constraints for better profitability.
For this assignment, you will test your understanding by constructing your LP model in Excel, carefully labeling all cells for clarity. Your submission will include the full spreadsheet showing your model setup and solution, along with a concise writeup explaining the problem, your model, the results, and the significance of the sensitivity analysis and shadow prices. This approach not only demonstrates your technical skills but also your ability to interpret and communicate the implications of LP solutions in real-world contexts.
References
- Bertsimas, D., & Tsitsiklis, J. N. (1997). Introduction to Linear Optimization. Athena Scientific.