Determine the daily production program that maximizes the company's output

Question

The problem:

The mechanical workshop can produce $600$ units of part #1 or $1200$ units of part #2 per shift. The production capacity of the thermal workshop, where these parts are sent for heat treatment on the same day, allows for processing $1200$ units of part #1 or $800$ units of part #2 per shift. The prices of both parts are the same. It is necessary to determine the daily production program of parts that maximizes the company's output, given the additional condition that both workshops work one shift.

---

So, in the end we obtain a mathematical model of the given problem in the form of a linear programming problem, both the objective function and the constraint functions should be linear. Which can be solved using simplex method.

Simply speaking, I need the problem above to be described in linear equation with constraints.

Example (not related to the problem above):

Answer 1

I assume that both workshops can produce both products in a shift and the working rate is constant. That is, e.g., the mechanical workshop could produce 300 of product 1 and 600 of product 2 in a single shift. A linear program could then look as follows:

Let $x_{i}$ be the units produced of product $i$ and $p_i$ the profit earned when producing (and selling) product $i$.

Linear Program \begin{align} \ & \max z = p_1 \cdot x_{1} + p_2 \cdot x_{2} \; \\ \\\textit{S.t.}\\ \\ & \frac{x_1}{600} + \frac{x_2}{1200} \leq 1 & \tag1 \\ \\ & \frac{x_1}{1200} + \frac{x_2}{800} \leq 1 & \tag2 \\ \\ & x_{1}, x_{2} \in \mathbb{N} \tag3 \end{align}

The objective is to maximize profit. Constraints (1) and (2) ensure that the capacities of workshops are not exceeded. Constraint (3) states that the units produced must be natural numbers.