Assignment 1: Linear Programming Case Study
Assignment 1 Linear Programming Case Studyyour Instructor Will Assign
Assignment 1. Linear Programming Case Study 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. Writeup. Your writeup should introduce your solution to the project by describing the problem. Correctly identify what type of problem this is.
For example, you should note if the problem is a maximization or minimization problem, as well as identify the resources that constrain the solution. Identify each variable and explain the criteria involved in setting up the model. This should be encapsulated in one (1) or two (2) succinct paragraphs. After the introductory paragraph, write out the L.P. model for the problem. Include the objective function and all constraints, including any non-negativity constraints.
Then, you should present the optimal solution, based on your work in Excel. Explain what the results mean. Finally, write a paragraph addressing the part of the problem pertaining to sensitivity analysis and shadow price. Excel. As previously noted, please set up your problem in Excel and find the solution using Solver.
Clearly label the cells in your spreadsheet. You will turn in the entire spreadsheet, showing the setup of the model, and the results.
Paper For Above instruction
Linear programming (LP) is a powerful mathematical technique used for resource allocation problems where decisions need to be optimized within certain constraints. In this case study, the problem involves determining the optimal production levels of two products, Product A and Product B, subject to constraints such as resource availability, labor hours, and raw material limits. The goal is either to maximize profit or minimize costs, making it a maximization or minimization problem, respectively. For this scenario, assume the problem aims to maximize profit by efficiently allocating resources.
The problem features at least three constraints, including resource limitations such as raw material supply, labor hours, and machine capacity. Additionally, it involves at least two decision variables representing quantities of products to produce, say, x₁ for Product A and x₂ for Product B. The decision variables are constrained to be non-negative, as negative production quantities are infeasible. The model's main objective is to maximize total profit, which is a linear function of the decision variables, with coefficients representing profit per unit for each product.
The LP model can be mathematically expressed as follows:
- Objective function: Maximize Z = c₁x₁ + c₂x₂
- Constraints:
- a₁₁x₁ + a₁₂x₂ ≤ b₁ (Resource 1 constraint)
- a₂₁x₁ + a₂₂x₂ ≤ b₂ (Resource 2 constraint)
- a₃₁x₁ + a₃₂x₂ ≤ b₃ (Resource 3 constraint)
- Non-negativity:
- x₁, x₂ ≥ 0
In this setup, the coefficients c₁ and c₂ represent the profit contributions per unit, while the a's and b's depict resource consumption and availability. After formulating the model, Excel's Solver can be used to identify the optimal production quantities that maximize profit while satisfying all constraints.
The optimal solution obtained through Solver provides the exact production quantities for Product A and Product B that maximize profit. This solution indicates how resources should be allocated to achieve the most profitable outcome, given the constraints. For instance, if Solver suggests producing 300 units of Product A and 150 units of Product B, this allocation maximizes profit without violating any resource constraints. The interpretation emphasizes the significance of decision variables' levels in relation to resource limitations and overall profitability.
A vital component of LP analysis involves sensitivity analysis and shadow prices. Sensitivity analysis examines how changes in resource availability or profit coefficients influence the optimal solution. In this context, the shadow price associated with a constraint indicates the potential increase in profit if the resource availability increases by one unit. A positive shadow price suggests that additional resource units could improve profit, whereas a zero shadow price implies that the resource is not a limiting factor at the optimal solution. Understanding these metrics assists managers in making informed decisions about resource investments and potential adjustments to constraints.
In conclusion, this LP model demonstrates how mathematical optimization can guide resource allocation decisions to maximize profits efficiently. Using Excel's Solver, the decision variables are optimized within the feasible solution space defined by constraints. The sensitivity analysis and shadow prices further provide insights into the robustness of the solution and the value of resources, thereby enabling strategic planning and resource management aligned with organizational goals.
References
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