A company supplies $5$ types of products to a retailer. It is contractually obligated to meet the demands of the retailer. Because of limited capacity the company may not have sufficient machining time, assembly time, and finishing time available to satisfy the entire demand through its in$-$house production. Contractual obligation requires the company to make up any shortfall by procuring the products, at higher costs, from an outsourcing partner. The company aims to meet the demands at minimum cost. The table below summarizes the relevant data. All costs are in dollars. The resource requirements for production are specified in minutes per unit. For example, it takes $1$ minute of machine time, $1$ minute of assembly time, and $2$ minutes of finishing time to produce each unit of P$1.$

Product P$1$ P$2$ P$3$ P$4$ P$5$

Demand $60004000100020009000$

Production cost $278310111661$

Purchase cost $359911613471$

Available hours Cost $/$ hour

Machine time $11311320240$

Assembly time $13344866180$

Finishing time $24241626300$

You must formulate the problem as a linear programming model, solve the model using available data to find an optimal production plan, and identify ways to improve the production process through sensitivity analysis.