# Model linearly: Determine amount of units for production

A company produces 2 products in a week. Let $x_i$ denote the number of units of product $i$ to produce. Each product requires liters of Chemical X to make.

Info is given below:

\begin{array}{|c|c|} \hline & & Prod \ 1 & Prod \ 2 \\ \hline \text{} & \text{ }& &\\ \hline \text{Production Hours/Unit} & & 4& 3 \\ \hline \text{Revenue/Unit} & & 100 &150\\ \hline \text{Liters of Chemical X/Unit} && 2 & 1 \\ \hline \end{array}

Others:

1. Company has to buy at least 50 L of Chemical X. It is prohibitively expensive to dispose Chemical X. Hence, it must use as much Chemical X as possible.

2. Company can produce at most 75 units total.

Company has goals:

1. Estimated Revenue of at least 7000/week

2. Produce at least 40 units of each product (for the upcoming week).

3. Use at most 120 hours of labour/week.

I have to formulate as a goal programming problem.

My formulation

Company has goals:

Goal 1: Minimum Revenue of 7000. $$100x_1 + 150x_2 \ge 7000$$

$$\to 100x_1 + 150x_2 + d_1^- - d_1^+ = 7000$$

$\$

Goal 2: 40 units each. $$x_1 \ge 40$$ $$x_2 \ge 40$$

$$\to x_1 + d_2^- - d_2^+ = 40$$ $$\to x_2 + d_3^- - d_3^+ = 40$$

$\$

Goal 3: At most 120 hours of labour. $$4x_1 + 3x_2 \le 120$$

$$\to 4x_1 + 3x_2 + d_4^- - d_4^+ = 120$$

$\$

Company has other constraints:

1. At least 50L of Chemical X should be produced. $$2x_1 + x_2 \ge 50$$

2. At most 75 units can be produced. $$x_1 + x_2 \le 75$$

3. Nonnegativity $$x_1, x_2, d_1^{\pm}, ..., d_4^{\pm} \ge 0$$

Is that right? In which part does or should the 'prohibitively expensive to dispose' come? I was thinking to include deviational variables in the 50L constraint but there is no goal for that.

Maybe $d_2^+$ and $d_3^+$ have their own constraints?