A publisher has orders for 600 copies of a certain text from San Franciso and 400 copies from Sacramento. The company has 700 copies in a warehouse in Novato, and 800 copies in a warehouse in Lodi. It costs 5 dollars to ship a text from Novato to San Francisco, but it costs 10 dollars to ship it to Sacramento. It costs 15 dollars to ship a text from Lodi to San Francisco, but it costs 4 dollars to shop from Lodi to Sacramento. How many copies should the company ship from each warehouse to San Francisco and Sacramento to fill the order at the least cost?
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Let $x_{ij}$ the amount of copies, which are shipped from warehouse $i$ to city $j$.
$i=1, \ \texttt{ if copies are transported from Navato}$
$i=2, \ \texttt{ if copies are transported from Lodi}$
$j=1, \ \texttt{ if copies are transported to San Francisco}$
$j=2, \ \texttt{ if copies are transported to Sacramento}$
First constraint: 600 copies are needed in San Francisco
$x_{11}+x_{21}=600$
Second constraint: 400 copies are needed in Sacramento
$x_{12}+x_{22}=400$
Third constraint: At most 700 copies can be shipped from warehouse in Navato
$x_{11}+x_{12}\leq 700$
Fourth constraint: At most 800 copies can be shipped from warehouse in Lodi
$x_{21}+x_{22}\leq 800$
The total costs of shipping have to be minimized:
$\texttt{min} \ \ 5 \cdot x_{11}+10\cdot x_{12}+15\cdot x_{21}+4\cdot x_{22}$
And finally the non-negative constraint: $ \ x_{ij}\geq 0 \quad \forall i,j$
If the solution is not an integer, don´t worry about it. Just round the results.
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