An entrepreneur has a window and door factory, which is why he has two production sections: assembly and finishing. The assembled window takes 1 hour and 2 hours to finish, while the assembled door takes 3 hours and 1 hour to finish. In the assembly section you can use a maximum of 9 hours and in the finishing section 8 hours a day. The businessman is aware that to have the maximum utility the door must be worth twice that of the window. How many windows and doors should be manufactured?

Let's assume that the entrepreneur produces x windows and y doors.

In the assembly section, it takes 1 hour to assemble a window and 3 hours to assemble a door. So the total time spent in the assembly section for windows and doors can be represented as:

$$1x + 3y$$

In the finishing section, it takes 2 hours to finish a window and 1 hour to finish a door. So the total time spent in the finishing section for windows and doors can be represented as:

$$2x + 1y$$

The constraint for the assembly section is that it should not exceed 9 hours, and for the finishing section, it should not exceed 8 hours. So we can write the following inequalities:

$$1x + 3y <= 9$$
$$2x + 1y <= 8$$

The entrepreneur wants the door to be worth twice that of the window. Mathematically, this can be represented as:

$$2x = y$$

Now, we need to find the values of x and y that satisfy all the above equations and inequalities.

To do that, we can solve these equations using the method of substitution or elimination.

Let's use substitution method here.

From the equation $2x = y$, we can substitute y in the other equations:

$$1x + 3(2x) <= 9$$
$$2x + 1(2x) <= 8$$

Simplifying these equations, we get:

$$7x <= 9$$
$$4x <= 8$$

Dividing both sides of the inequalities by their respective coefficients, we get:

$$x <= \frac{9}{7}$$
$$x <= 2$$

Since x represents the number of windows, it cannot be a fraction or negative. So the maximum possible value for x is 2.

Substituting this value of x in the equation $2x = y$, we get:

$$2(2) = y$$
$$y = 4$$

So, the maximum number of windows that can be manufactured is 2, and the maximum number of doors that can be manufactured is 4.

Answer: The entrepreneur should manufacture 2 windows and 4 doors.