Maximize Z$=4$x$\_(1)+8$x$\_(2)+10$x$\_(3)+6$x$\_(4)$

subject to

x$\_(1)+$x$\_(2),+$x$\_(5)+$xconstraint$(1)/($ r$)$esource $1)$

x$\_(2)+$x$\_(3),+$x$\_(6)+$xconstraint$(2)/($ r$)$esource $2)$

x$\_(3)+$x$\_(4),+$x$\_(7)+$xconstraint$(3)/(\_(\_()))$ resource $3)$

x$\_(1),-2$xconstraint$(4)/($ r$)$esource $4)\_()$

x$\_(2),-2$xconstraint$(5)/($ r$)$esource $5)$

x$\_(3),-2$xconstraint$(6)/($ r$)$esource $6)\_()$

x$\_(4),-2$xconstraint$($ Z$)/($ r$)$esource $7)\_()$

and x$\_($j$)>=0,$j$=1,2,$cdots,$8.$

$($a$)$ Make up a story that has this linear programming model.

$($b$)$ Use Excel Solver to find the optimal solution.

$($c$)$ Identify which constraint is binding.

$($d$)$ Use Excel Solver to find shadow price for each resource. $($Do not use ASPE's solver, which was

not covered in lecture anyway$)$

$($e$)$ Use the simplex method in tabular form to find the optimal solution step by step.

Because this involves many iterations $($about $8$ iterations$),$ I am giving you the simplex tableau after the

$4^($th $)$ iteration for your reference. If there are multiple rows with the same minimum ratio, let's choose the

first min ratio row, that way we should all end up with the same tableau after each iteration. For

example, if Eq$2$ row and Eq$3$ row both have the same minimum ratio, we choose Eq$2$ row as the pivot

row.

$($f$)$ Use the last tableau of the simplex method to read shadow price for each resource.