Problem $8$:

Given the following linear programming results for a cost minimization problem. In this problem, the choice variables $)$ are inputs; the firm must choose the optimal amount of each input to keep total costs at a minimum. The constraints $(b_i)$ represent production $($output$)$ of three products shirts, pants, and shoes.

$\backslash$table$[[,A,B,C,D,E,F,$G$,H,1],[3,,x_1,x_2,x_3,x_4,,,$TC$,],[4,$Input,$0\mathrm{.}00,5\mathrm{.}94,0\mathrm{.}00,7\mathrm{.}50,,,101\mathrm{.}25,],[5,$Per Unit Input Cost,$2,12,5,4,,,,],[6,,,,,,,,,],[7,,x_1,x_2,x_3,x_4,,,,],[8,$Constraints,Resource Usage,Resource Usage,Total LHS$,,$RHS$,$Slack$],[9,$Output $1($Shirts$),1,8,2,3,,,70,0\mathrm{.}00],[10,$Output $2($Pants$),8,6,4,5,73\mathrm{.}13,,40,33\mathrm{.}13],[11,$Output $3($Shoes$),2,8,4,7,100\mathrm{.}00,,100,0\mathrm{.}00]]$

Variable Cells

$\backslash$table$[[$Cell$,$Name,$\backslash$table$[[$Final$],[$Value$]],\backslash$table$[[$Reduced$],[$Cost$]],\backslash$table$[[$Objective$],[$Coefficient$]],\backslash$table$[[$Allowable$],[$Increase$]],\backslash$table$[[$Allowable$],[$Decrease$]]],[$$B$$4,$Input $1,0,0\mathrm{.}625,2,1E+30,0\mathrm{.}625],[$$C$$4,$Input $2,5\mathrm{.}9375,0,12,20,1\mathrm{.}333333333],[$$D$$4,$Input $3,0,2\mathrm{.}25,5,1E+30,2\mathrm{.}25],[$$E$$4,$Input $4,7\mathrm{.}5,0,4,0\mathrm{.}5,1E+30$