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In an interview, I was asked to frame the following optimization problem:

  1. A company produces two products $x,y$ using two machines $a,b$.
  2. To produce each product, both the machines are used together. It is not like either of the machines can be used to produce a product.
  3. Producing 1 unit of product $x$, it requires to run machine $a$ for 50 mins and machine $b$ for 30 mins.
  4. Producing 1 unit of product $y$, it requires to run machine $a$ for 24 mins and machine $b$ for 33 mins.
  5. Stock in hand for product $x$ is 30, $y$ is 90.
  6. Demand for product $x$ is 75 units, product $y$ is 95 units.
  7. Machine $a$ can be run at max for 40 hours, and $b$ can be run at max for 35 hours.
  8. The company wants to maximize the total production units of $x$ & $y$ ensuring it meets the demand.

The whole scenario is just for 1 week.

I framed it in following way. I chose the decision variables $x$ and $y$ representing units of product $x$ and $y$, respectively.

$$ \begin{array}{ll} \underset{x,y}{\text{maximize}} & x + y \\ \text{subject to} & 50x + 24y \leq 40 \cdot 60 \\ & 30x + 33y \leq 35 \cdot 60 \\ & x \geq 75-30 \\ & y \geq 95-90 \end{array} $$

I missed to add the integer constraints on $x$ & $y$ (they represent unit sales).

There were no more follow up questions, and I was rejected on the ground of technical depth. Can anyone help if I made a blunder in the above attempt to model the problem?

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    $\begingroup$ I think what you have looks fine $\endgroup$ Commented Mar 30 at 14:17
  • $\begingroup$ There could be more angles added to it - should we minimize machine runtime instead, or, should we allow machines to run beyond the limit to meet demand, etc. Do you think it is just missing a soft constraint, e.g. on machine run time extension? $\endgroup$ Commented Mar 30 at 14:19
  • $\begingroup$ In my experience, optimization problem formulation using mathematical modeling is an iterative process. If the initial solution does not make sense, probably due to missing information, then updates are made to improve and make it realistic. Only God knows what the interviewer wanted to hear, they did not have any questions! :D $\endgroup$ Commented Mar 30 at 17:41

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Besides the integrality restrictions, there are a couple of aspects you might have asked about.

  • One is whether they value production of the two products equally. Your objective implicitly assumes that a unit of one product has the same value to management as a unit of the other product. Maybe this is true, maybe not.
  • Another is whether they are content to end the planning period with no inventory of one or the other product (which is possible given your formulation, although I suspect not likely) or whether there is some minimum inventory level with which they would be comfortable. They might want the right-hand sides of the last two constraints tweaked so that they don't start the next period with empty shelves.

Personally, I would not think either of those would be an indication of lack of technical depth.

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  • $\begingroup$ Thank you. I asked for the first point, but not the second one. I think they were expecting me to ask these questions upfront. I assumed it would be like a conversation, hence, I first shared my approach and expected follow-up questions, that didn't happen. Nevertheless, some learnings - "keep asking" :) $\endgroup$ Commented Mar 30 at 17:52

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