Solve the following case study using the transportation method:

Suppose there is a manufacturing company that produces and distributes its products to various customer locations across the country. This company has three production plants $($Plant A$,$ Plant B$,$ and Plant C$)$ located in different locations, and there are also five distribution centers $($DC $1,$ DC $2,$ DC $3,$ DC $4,$ and DC $5)$ scattered across various regions.

The company's goal is to determine the most efficient transportation routes for delivering products from the plants to the distribution centers, considering transportation costs and available capacities.

$\backslash$table$[[$Sources$,$Distribution Center,,$],[$DC $1,$DC $2,$DC $3,$DC $4,$Capacity$],[$Plant A$,$$$145,$$$150,$$$165,$$$140,2000],[$Plant B$,$$$175,$$$135,$$$120,$$$165,1250],[$Plant C$,$$$113,$$$156,$$$134,$$$148,1250],[$Demand$,1000,1200,1400,900,]]$