Problem Description:

In this project for the ENIE$211$ Operations Research class, you will be addressing a real$-$world problem related to optimizing the reorganization of the dollar using linear programming techniques. The objective is to assign numerical values to each denomination of the dollar in a way that optimizes resource allocation and achieves a specific goal.

Objective:

Define the objective of the linear programming model. Clearly state the goal you aim to achieve, such as maximizing profit, minimizing cost, or optimizing resource allocation.

Assumptions:

$1.$ Availability of all denominations: Assume that all denominations of the dollar are available for reorganization, including $$1,$ $$5,$ $$10,$ $$20,$ $$50,$ and $$100$ bills.

$2.$ Fixed denominations: Assume that the values of the denominations remain fixed and cannot be changed during the reorganization process.

$3.$ Whole numbers: Assume that the number of bills used for each denomination must be a whole number, as fractional bills cannot be used.

$4.$ Non$-$negativity: Assume that the number of bills used for each denomination cannot be negative.

Variables:

Define the decision variables for the problem. Clearly specify what each variable represents in the context of the problem and how they contribute to achieving the objective. For example:

$-$ x$1$ represents the number of $$1$ bills used.

$-$ x$5$ represents the number of $$5$ bills used.

$-$ x$10$ represents the number of $$10$ bills used.

$-$ x$20$ represents the number of $$20$ bills used.

$-$ x$50$ represents the number of $$50$ bills used.

$-$ x$100$ represents the number of $$100$ bills used.

Constraints:

Identify and describe the constraints that need to be considered in the model. Ensure that each constraint is directly related to the problem's conditions and limitations$.$ For example:

$-$ The total value of bills used should be equal to the desired dollar amount.

$-$ The total number of bills used should be minimized.

Objective Function:

Formulate the objective function that needs to be optimized. Clearly indicate whether it is a maximization or minimization problem. Provide a mathematical representation of the objective function. For example:

Minimize: x$1+$ x$5+$ x$10+$ x$20+$ x$50+$ x$100$

Model Formulation:

Present the complete linear programming model, including the objective function and all constraints, in a mathematical format. Ensure that the formulation accurately represents the problem and its requirements.

Sensitivity Analysis:

Discuss the importance of performing sensitivity analysis on the solution. Explain how changes in the coefficients of the objective function or constraints may impact the solution and its validity.

Conclusion:

Summarize the findings of your analysis and the insights gained from solving the problem. Reflect on how the developed linear programming model effectively addresses the problem at hand. Additionally, discuss any limitations or potential improvements that could enhance the model's performance.

References:

Cite all the sources and tools used in the project to acknowledge the relevant information and methodologies utilized.

Important Dates:

Submission Deadline: May $15$

Note: Please ensure that your group members remain the same as originally assigned earlier this semester. This project is for the ENIE$211$ Operations Research class in the Department of Industrial Engineering.