A smartphone manufacturing company operates three factories, each situated in one of three cities: New York, San Francisco, and Los Angeles. These factories supply the manufactured smartphones to three distribution centers located in the city of Chicago. The per-unit transportation cost matrix is provided below:

The factory in New York has a capacity of 15,000 units.

The factory in San Francisco has a capacity of 18,000 units.

The factory in Los Angeles has a capacity of 8,000 units.

The requirements of the distribution centers are:

1. Determine how much of the companys production should be shipped from each factory to each distribution center to minimize the total transportation cost.
2. What would be the effect on the optimal solution and the optimal cost if the demand at DC3 increases to 15,000 units?
3. Suppose a new distribution center (DC4) can be used that requires 5,000 units. The transportation cost from New York to DC4 is $4 per unit, from San Francisco to DC4 is $3 per unit, and from Los Angeles to DC4 is $5 per unit. Does using this new distribution center beneficial for the company? Explain from transportation cost and revenue perspectives.

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\begin{tabular}{|c|c|c|c|} \hline \multicolumn{4}{|c|}{ Transportation Costs (\$) } \\ \hline Factories/Distribution Center (DC) & DC1 & DC2 & DC3 \\ \hline New York & 4 & 3 & 7 \\ \hline San Francisco & 7 & 6 & 4 \\ \hline Los Angeles & 3 & 6 & 6 \\ \hline \end{tabular} \begin{tabular}{|c|c|} \hline Distribution Centers & Requirement (Smartphones) \\ \hline DC1 & 18,000 \\ \hline DC2 & 12,000 \\ \hline DC3 & 5,000 \\ \hline \end{tabular}