The Fagersta Steelworks currently is working two mines to obtain its iron ore. This iron ore is shipped to either of two storage facilities. When needed, it then is shipped on to the company's steel plant. The diagram below depicts this distribution network, where M$1$ and M$2$ are the two mines, S$1$ and S$2$ are the two storage facilities, and P is the steel plant. The diagram also shows the monthly amounts produced at the mines and needed at the plant, as well as the shipping cost and the maximum amount that can be shipped per month through each shipping lane. $($Go to the left column below the diagram.$)$

$($a$)$ Please formulate a linear programming model for this problem. $(1$ points$)$

$($b$)$ Solve this model by the simplex method, without any software, not excel or not other software, it should be manually.$|$ how to calculate the amount shipped? is there any formula or soemthing? please expalin how to get it$.$ Also, how to get the total cost.

$\backslash$table$[[,$M$1-$S$1,$M$1-$S$2,$M$2-$S$1,$M$2-$S$2,$S$1-$P$,$S$2-$P$,,,],[$Unit Cost,$$2,000,$$$1,700,$$$1,600,$$$1,100,$$$400,$$$800,,,],[,,,,,,,,$REQUIRED$],[$Contribution Toward Required Amount,TOTALS,,AMOUNT$],[$M$1,1,1,,,,,40,=,40],[$M$2,,,1,1,,,60,=,60],[$S$1,1,,1,,-1,,0,=,0],[$S$2,,1,,1,,-1,0,=,0],[P,,,,,1,1,100,=,100],[,,,,,,,,],[,,,,,,,,$TOTAL COST$],[$AMOUNT SHIPPED$],[,,,,,,,,,],[$CAPACITY$,30,30,50,50,70,70,,,]]$