A company has a machine to manufacture three different products P1, P2 and P3. Each unit of products P1, P 2 and P3 sold by the firm provides it with a profit of $4,$12 and $3, respectively. Furthermore, the production capacity of the machine for each of the three products is 50,25 and 75 units per hour. Finally, note that technological limitations mean that it is not possible to manufacture more than 1,000 units of product P1, 500 units of product P 2 and 1,500 units of product P 3 during a week. Assuming that the machine operates 45 hours per week, the problem is to determine the number of units to be produced in a week for each of the products P1,P2 and P3 so as to maximize profit. 1. Formulate this problem using a linear programming model using only integer coefficients. Do not forget to provide an interpretation of the decision variables, the objective and the constraints. 2. Explain how one can identify the optimal solution of this problem by simple reasoning (i.e., without using the simplex algorithm)