A car sequencing problem is an example of an industrial optimization problem. Cars are available with many different options. Car color, leather seats, wood interior, sunroof, type of motor, Global Positioning System (GPS), air conditioning, and so on are all examples of options. Cars can be configured with different sets of options. For example, a manufacturer will produce a certain number of blue cars with leather seats, wood interior, and air conditioning, a certain number of green cars with sunroof and GPS, and so on for each model of car.

The problem in car sequencing is to determine the order in which cars of each configuration should be assembled. The assembly line must produce a certain number of cars of each configuration. Not every car can be fitted with every option, due to technical limitations. For technical and financial reasons, the order in which cars are sent along the assembly line, or the sequence of the cars, is important.

The sunroof option is one example of how technical limitations restrict the number of cars with a certain option and restrict the order of the cars on the assembly line. As the car moves along the assembly line, a machine and crew move along with it, installing the sunroof. The machine can only move a certain distance along the assembly line. In the time it takes the machine to move along the assembly line with the car, the sunroof is installed. After the sunroof is installed, the machine moves back to its original position on the assembly line. At this stage, however, several cars have already passed the starting point of the sunroof machine and crew. Therefore, only the next car to pass the starting point can have the sunroof option. Those cars that have already passed cannot have it installed. It is important that cars that require a sunroof are in the right place at the right time or in the correct sequence. If an option becomes so popular that it needs to be installed on every car, then the assembly line is modified to reflect this.

The car paint color option is another example of how technical limitations restrict the number of cars with a certain option and restrict the order of the cars on the assembly line. The machine that paints the car must be serviced after every four cars. The paint is emptied, the tubes and pipes are cleaned, and a new color of paint is added. Only four cars can be painted a certain color before the machine is serviced. Then four cars can be painted another color before the machine is again serviced. The order of these cars is important to make sure that the color of the car is matched with the other options in a particular configuration.

Assume that you have eight cars to paint in three available colors: green, yellow, and blue. Due to technical limitations on the assembly line, no more than three cars can be painted green, exactly three cars must be painted yellow, and no more than two cars can be painted blue. Due to limitations on the order of the cars, the first car off the assembly line cannot be painted green.

The first step in modeling and solving a problem is to write a natural language description of the problem, identifying the decision variables and the constraints on these variables.

Write a natural language description of this problem. Answer these questions: