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Attempt

Let $x_1$ be the number of hours to produce product 1 during assembly and $x_2$ be the number of hours to produce product 1 during finishing. Notice that $\frac{1}{2} x_1 + x_2 $ is the number of product 1 units produced. Similarly, call $x_3$ and $x_4$ be the number of hours spent during assembly and finishing to produce product two. We see thtat the number of units of product 2 is $x_3 + 2 x_4$. We want to maximize profit. So, our objective function is

$$ z = 25 (0.5 x_1 + x_2) + 15 (x_3+2x_4) $$

subject to

$$ x_1 + x_3 \leq 120 $$

$$ x_2 + x_4 \leq 85 $$

$$ x_1,x_2,x_3,x_4 \geq 0 $$

To find a worth of an hour of assembly and an hour of finishing, do we need to find the dual solution? Since the dual varaibles (y_1,y_2) that corresponds to each constraint one measures hours of assembly and $y_2$ measures hours of finishing. But, what is the units of the dual objective? Am I on the right track?

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1 Answer 1

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I would use two variables, $z_1, z_2$ to be the number of chairs to be made

$$\max 25 z_1 + 15z_2$$

subject to

$$2z_1 + z_2 \le 120$$

$$z_1 + \frac12z_2 \le 85$$

$$z \ge 0$$

Notice that the finishing constraint is never active.

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  • $\begingroup$ Thanks for the reply. So the answer to this problem would then be just the dual variables correct ? $\endgroup$ Commented Nov 8, 2018 at 8:20
  • $\begingroup$ hmm... unlikely. It can't be that we are not paying someone at all. Salary negotiation in my interpretation is a "game", depending on who knows what and the market situation. We know that a single chair is earning us $\$15$ and that takes $1$ hour of assembly and $\frac12$ hour of finishing. $\endgroup$ Commented Nov 8, 2018 at 8:25
  • $\begingroup$ So how can we find then the worth of an hour assembly and finishing workers time ? $\endgroup$ Commented Nov 8, 2018 at 8:27
  • $\begingroup$ i do not know. I think most likely I misinterpreted the question. I keep viewing the question as a game rather than a single player problem. Is the question asking for shadow price? $\endgroup$ Commented Nov 8, 2018 at 8:30
  • $\begingroup$ The problem is on duality section of my notes, so I guess its related to shadow prices. $\endgroup$ Commented Nov 8, 2018 at 8:35

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