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A company makes two products, product 1 (X1) and product 2 (X2).

Profits per unit are $\$30.00$ for $X_1$ and $\$15.00$ for $X_2$.

Hours per unit for each of the three departments are:

Dept. A (hrs./unit) $1.00$ for $X_1$, $0.35$ for $X_2$ with a maximum capacity of $100$ hours for the whole department

Dept. B (hrs./unit) $0.30$ for $X_1$, $0.20$ for $X_2$ with a maximum capacity of $36$ hours for the whole department

Dept. C (hrs./unit) $0.20$ for $X_1$, $0.50$ for $X_2$ with a maximum capacity of $50$ hours for the whole department

There is a possibility to work extra hours: $10$ extra hours for Dept. A at $\$18$ per hour, 6 for Dept. B at $\$22$ per hour, and $8$ for Dept. C at $\$12$ per hour. However, there can only be 2 departments at a time with extra hours.

I formulated the problem as follows:

Let $X_i$ be the number of units of product $i$, $H_N$ the extra hours for department $N$, and $O_A$, $O_B$ and $O_C$ binary variables that represent which departments are working extra hours

Max $30X_1 + 15X_2$ subject to

$X1 + 0.35X_2 - H_A \leftarrow 100$

$0.3X1 + 0.2X_2 - H_B \leftarrow 36$

$0.2X_1 + 0.5X_2 -H_C \leftarrow 50$

$H_A \leftarrow 10(O_A)$

$H_B \leftarrow 6(O_B)$

$H_C \leftarrow 8(O_C)$

$O_C + O_B + O_A \leftarrow 2$

Is this right??

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  • $\begingroup$ The additional costs for each extra hour worked should be included in the objective function. $\endgroup$ – YukiJ Feb 18 at 8:26
  • $\begingroup$ Great! thank you! $\endgroup$ – Lollipop Feb 18 at 14:53

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