Consider the weekly production planning problem of an automobile manufacturer, who is producing two car models in its plant, which are car types 1 and 2. Each car requires certain steps in manufacturing to reach its end-user version, consisting of operations in body manufacturing process in welding department, coating department and assembly department, respectively. In particular,

Car type 1 requires 6 hours in welding department, 7 hours in coating department, and 3 hours in assembly department to be completed.

Car type 2 requires 7 hours in welding department, 3 hours in coating department, and 8 hours in assembly department to be completed.

Furthermore, each department has a certain labor force, resulting in weekly production capacities. Specifically, welding, coating and assembly departments have weekly capacities of 320, 170 and 360 hours, respectively.

Additionally, the unit profit of selling a car type 1 is 30000 TL and a car type 2 is 25000 TL, and sales department informs that each car type produced can be sold to the market. Next, the company needs to determine its weekly production schedule of cars to maximize its profit.

Combining the above, the company decides to model this problem as the below mathematical program:

where x₁ and x₂ represent the number of type 1 and 2 cars produced on a given week, respectively. Assume that production amounts of cars can take continuous values as given in the model, so do not revise them as integer variables.

The automobile manufacturer now notices that producing two car types in each week might result in some issues as each manufacturing department needs to take into account additional setup times as car type changes. To handle this issue, the manufacturer decides to produce one car type per each week. Revise the original mathematical model given in the problem under this change and write down your revised formulation clearly.