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I have some doubts about the approach to the constraints in this linear programming problem.

A company manufactures domestic fans. It currently markets fans in three different sizes: A, B and C which provide a net profit of €30, €36 and €42, respectively.

  • The company has 3 production plants, 1, 2 and 3 with a labour and equipment capacity to produce 90, 75 and 45 units per day of this product, respectively, regardless of the size or combination of sizes involved.
  • The amount of space available to store in-process material also imposes limitations on the production of the new product. Plants 1, 2 and 3 have 130, 100 and 70 square metres of space, respectively, for in-process material in daily production.
  • Each type of fan A, B and C produced requires 1.2, 1.5 and 2 square metres, respectively.
  • Sales forecasts indicate that, if available, no more than 75, 120 and 90 units per day of the respective models A, B and C can be sold.
  • It should be noted that at least 10 units of model A need to be produced to satisfy an outstanding order.
  • In addition, management has decided that all three plants have to use the same proportion of their labour capacity to manufacture the fans.

I have written the following table simply to be able to identify the variables of the problem and to identify the constraints in a simpler way.

\begin{array}{|c|c|c|} \hline \text{ } & P_{1} & P_{2} & P_{3} & \text{Max} & \text{Size} \\ \hline \text{A} & X_{1} & X_{2} & 75 - X_{1} -X_{2} & 75 & 1.2 \\ \hline \text{B} & X_{3} & X_{4} & 120 - X_{3} -X_{4} & 120 & 1.5 \\ \hline \text{C} & 90 - X_{1} - X_{3} & 75 - X_{2} - X_{4} & 45 - (75 - X_{1} -X_{2}) - (120 - X_{3} -X_{4}) & 90 & 2 \\ \hline \text{Production} & 90 & 75 & 45 \\ \hline \text{Space} & 130 & 100 & 70 \\ \hline \end{array}

I have proposed the following inequalities:

$ X_{1}, X_{2}, X_{3}, X_{4} \geqslant 0 \\[10pt] \text{Production constraints:} \\[10pt] 90 - X_{1} - X_{3} \geqslant 0 \text{ }\to\text{ } X_{1} + X_{3} \leqslant 90 \\ 75 - X_{2} - X_{4} \geqslant 0 \text{ }\to\text{ } X_{2} + X_{4} \leqslant 75 \\ 45 - (75 - X_{1} -X_{2}) - (120 - X_{3} -X_{4}) \geqslant 0 \text{ }\to\text{ } X_{1} + X_{2} + X_{3} + X_{4} \geqslant 150 \\[10pt] \text{Amount of space:} \\[10pt] 1.2X_{1} + 1.5X_{3} + 2(90 - X_{1} - X_{3}) \leqslant 130 \text{ }\to\text{ } 0.8X_{1} + 0.5X_{3} \geqslant 50 \\ 1.2X_{2} + 1.5X_{4} + 2(75 - X_{2} - X_{4}) \leqslant 100 \text{ }\to\text{ } 0.8X_{2} + 0.5X_{4} \geqslant 50 \\ 1.2(75 - X_{1} - X_{2}) + 1.5(120 - X_{3} -X_{4}) + 2(-150 + X_{1} + X_{2} + X_{3} + X_{4}) \leqslant 70 \text{ }\to\text{ } 0.8X_{1} + 0.8X_{2} + 0.5X_{3} + 0.5X_{4} \leqslant 100 $

The truth is that I am not entirely sure that I have got the restrictions right. I have several doubts, for example:

  • Have I written enough (and minimum) variables to solve the problem?
  • The restriction on the minimum order of A does not make sense to me since I think it will be fulfilled as a result of fulfilling some of the previous restrictions.
  • The constraint on the proportion of fans produced in the three plants is also missing because I think that the previous constraints ensure that this will be the case.

Does anyone notice any errors in the inequations?

Thank you very much in advance.

cooper
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    In my opion you should work with the follwoing variables: $x_i$=amount of fan type $i$ and $x_{ij}$=amount of fan type $i$ produced in plant $j$, where $\sum\limits_{j=1}^3 x_{ij}=x_i$ So my main advice is to work with double indexed variables. – callculus42 Mar 23 '21 at 09:51
  • What do you think of my suggestion? – callculus42 Mar 23 '21 at 19:11
  • @callculus, sorry for the delay in replying. Your suggestion was a great help to me. I have been able to approach the problem in a much simpler way. Thank you very much! – cooper Mar 23 '21 at 21:21
  • It pleases me that the comment has helped. – callculus42 Mar 24 '21 at 04:33

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