Question 1. Linear Programming Models [9 Points]

A company supplies 4 types of products to a retailer. It is contractually obliged 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 for 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 3 minutes of machine time, 2 minutes of assembly time, and 3 minutes of finishing time to produce each unit of P1.

Product	P1	P2	P3	P4
Demand	6000	7000	4000	8000
Production cost	47	67	89	33
Purchase cost	55	80	93	39
					Available hours	Cost / hour
Machine time	3	4	1	1	928	120
Assembly time	2	4	1	2	960	240
Finishing time	3	3	3	3	1200	60

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.

Task 1.1. Formulate the problem as a linear programming model. [2 Points]

Decision variables:

Objective Function:

Subject to Constraints: