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It is given the following transportation problem. I can solve it when the cost is increased, but I don't understand what is the change in the model when the transport cost is reduced.

Problem: It is required to move machines from factories $A, B$ and $C$ to warehouses $X, Y$ and $Z$. There are $5$ required at $X$, $4$ at $Y$ and $3$ at $Z$, whilst there are $8$ available at $A$, $5$ at $B$ and $3$ at $C$. The transport cost in £ between the sites is given in the following table:

Transport cost: $$\begin{array} {|r|r|}\hline & X & Y & Z \\ \hline A & 50 & 60 & 30 \\ \hline B & 60 & 40 & 20 \\ \hline C & 40 & 70 & 30 \\ \hline \end{array}$$

Suppose now there is a reduction in the transport cost of $£10$ per machine taken from A for any machines above six. Find the new optimal solution, but because this time there has been a decrease in price for higher quantity, you will need to solve two problems and also use pricing out.

Thank you for any suggestions.

xrfxlp
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Kleona
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  • Why is the cost matrix $3\times 4$ for $3$ factories and $3$ warehouses? – Marcus Ritt Jun 19 '19 at 19:13
  • It was the wrong table. I edited it. Thank you. – Kleona Jun 20 '19 at 09:51
  • Note that the modified problem is no longer really a linear programming problem since the objective function is piecewise-defined. The hint is to solve two problems (one for each version of the objective function, not worrying at first about the cutoff point) and then use shadow prices to decide what your ultimate decision will be (since e.g. you might need to increase the amount sent from A to actually take advantage of the reduced price, moving the solution from one of the two solutions you just computed. If this moved solution is better than the other solution, take it). – John Coleman Jun 20 '19 at 11:45
  • How can you move machine from factory to warehouses when there are more machines than warehouses can have? – xrfxlp Jun 20 '19 at 12:09
  • @JohnColeman thank you. Your explanation really helped me. – Kleona Jun 22 '19 at 20:25

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