Chem Labs uses raw materials I and II to produce two domestic cleaning solutions, A and B. The daily availabilities of raw materials I and II are 150 and 145 units, respectively. One unit of solution A consumes 0.5 unit of raw material I and 0.6 unit of raw material II, and one unit of solution B uses 0.5 unit of raw material I and 0.4 unit of raw material II. The profits per unit of solutions A and B are 8 and $10, respectively. The daily demand for solution A lies between 30 and 150 units, and that for solution B between 40 and 200 units. Find the optimal production amounts of A and B.

My attempt

Let A and B be the no. of units of A and B produced and X and Y be no. of raw materials I and II to be processed respectively.

The objective function is to maximize the profit, Z.


The objective function is subject to the following constraints





Is this formulation correct? If it is, how can one proceed from this point to find the maximum profit?


The use of raw materials $I$ and $II$ depends on the production of $A$ and $B$, so you don't need the variables $X$ and $Y$.

You can create a table of given data:

$$\begin{array}{c|c|c|c} Products&I&II&Profit&Demand\\ \hline A&0.5&0.6&8&30\le A\le 150\\ B&0.5&0.4&10&40\le B\le 200\\ \hline Available&\le 150&\le 145&maximize&\end{array}$$

Now we can formulate the LPP: let $A$ and $B$ be the numbers of units of $A$ and $B$, respectively. Then: $$\pi(A,B)=8A+10B\to \text{max} \ \ \text{subject to}\\ 0.5A+0.5B\le 150 \ \ \text{(material I constraint)}\\ 0.6A+0.4B\le 145 \ \ \text{(material II constraint)}\\ 30\le A\le 150 \ \ \text{(demand for A)}\\ 40\le B\le 200 \ \ \text{(demand for B)}\\ $$ You can use graphical or Simplex methods to solve LPP.

Graphical method.

1) Draw the feasible (green) region from the constraint inequalities:

enter image description here

2) Find the corner points: $A,B,C,D,E,F$.

3) Evaluate the objective (profit) function at the corner points and choose the maximum.

Can you do it?


$\pi(100,200)=2800.$ WolframAlpha answer.


We just have to decide how many units of $A$ and $B$ are to be produced.

You are right that the profit is $8A+10B$ and we want to maximize it.

Now, let's examine the constraint imposed by material I.

$$0.5A+0.5B \le 150$$

Now, let's examine the constraint imposed by material II. $$0.6A + 0.4B \le 145$$

The demand informations also gives us

$$30 \le A \le 150$$


$$40 \le B\le 200.$$

Now, we have a $2$-dimensional linear programming problem, you can use a graphical method to solve the problem if you wish.


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