To solve this optimization problem, we need to maximize the profit while considering the constraints on raw materials and demand. This is a linear programming (LP) problem, where:

• Objective: Maximize profit by choosing optimal amounts of solutions $A$ and $B$ to produce.
• Constraints: Availability of raw materials, demand for solutions $A$ and $B$, and non-negativity of production amounts.

Step 1: Define the decision variables

Let:

• $x_1$ = number of units of solution $A$ produced.
• $x_2$ = number of units of solution $B$ produced.

Step 2: Write the objective function

The total profit $P$ is given by: $P = 8x_1 + 10x_2$ This is what we want to maximize.

Step 3: Define the constraints

Raw material availability constraints:

• For raw material $/$: $0.5x_1 + 0.5x_2 \leq 150$
• For raw material $//$: $0.6x_1 + 0.4x_2 \leq 145$

Demand constraints:

• The demand for solution $A$ lies between 30 and 150 units: $30 \leq x_1 \leq 150$
• The demand for solution $B$ lies between 40 and 200 units: $40 \leq x_2 \leq 200$

Non-negativity constraints:

$x_1 \geq 0, \quad x_2 \geq 0$

Step 4: Linear programming formulation

The problem can now be formulated as follows:

Maximize: $P = 8x_1 + 10x_2$

Subject to:

1. $0.5x_1 + 0.5x_2 \leq 150$ (raw material / constraint)
2. $0.6x_1 + 0.4x_2 \leq 145$ (raw material // constraint)
3. $30 \leq x_1 \leq 150$ (demand for A)
4. $40 \leq x_2 \leq 200$ (demand for B)
5. $x_1 \geq 0$, $x_2 \geq 0$ (non-negativity constraints)

Step 5: Solve using linear programming

To find the optimal solution, we can use a method such as the Simplex Method or a solver for linear programming. I will now calculate the optimal solution for this problem.The optimal production amounts for ChemLabs are:

• Produce 100 units of solution $A$.
• Produce 200 units of solution $B$.

This results in a maximum profit of $2800 per day.

Would you like further details or have any other questions? Here are some related questions to explore further:

1. How would the solution change if the availability of raw material $//$ increased to 160 units?
2. What happens to the profit if the profit per unit of solution $B$ decreases to $9?
3. Can you explain the Simplex Method used in linear programming?
4. What if the demand for solution $B$ decreases to a range of 30-100 units?
5. How would the solution be affected if both raw materials were more expensive?

Tip: Linear programming is useful for optimizing problems with constraints; graphical methods work for two variables, but solvers like Simplex handle larger cases efficiently.