$(20$ Points$)$ You are working for the IT department at OSU Lima and in charge of maintaining printing machines on campus. Due to high cost of brand$-$name printer cartridges, your boss asked you to research four secondary manufactures $-$ A$,$ B$,$ C$,$ and D $-$ and recommend two of them for supply of printer cartridges. The prices are equivalent, so you only need to make a decision on the number of pages printed before a cartridge runs out. So you bought $4$ new cartidges from each manufacturer and installed them in $16$ HP M$603$ printers across the campus. The data you collected are the number of pages $x_{ij}$ that were printed by cartridge i made by supplier $j.$ Below is summarized data.