Problem $2($Each question is worth two marks. Total sixteen marks$)$

Maximize $15$x$1+20$x$2+14$x$3$ where

x$1=$ quantity of product $1$

x$2=$ quantity of product $2$

x$3=$ quantity of product $3$

subject to

Labor $5$x$1+6$ x$2+4$x$3<=210$ hours

Material $10$x$1+8$ x$2+5$x$3<=200$ pounds

Machine $4$x$1+2$x$2+5$x$3<=170$ minutes

x$1,$ x$2,$ x$3>=0$

The following information was obtained from the output. The ranges were also computed based

on the output, and they are shown as well.

Total profit $=548\mathrm{.}00$

Variable Value Reduced Cost Range

Product $1010\mathrm{.}6$ Unlimited to $25\mathrm{.}60$

Product $25209\mathrm{.}40$ to $22\mathrm{.}40$

Product $332012\mathrm{.}50$ t$050\mathrm{.}00$

Constraint Slack Shadow Price Range of Feasibility

Labor $520\mathrm{.}0158\mathrm{.}00$ to unlimited

Material $02\mathrm{.}4170\mathrm{.}00$ to $270\mathrm{.}91$

Machine $00\mathrm{.}450\mathrm{.}0$ o $200\mathrm{.}00$

a$.$ Which decision variables are basic $($i$.$e$.,$ in solution$)?$

b$.$ By how much would the profit per unit of product $1$ have to increase for it to have a

nonzero value $($i$.$e$.,$ for it to become a basic variable$)?$

c$.$ If the profit per unit of product $2$ increased by $$2$ to $$22,$ would the optimal production

quantities of products $2$ and $3$ change? Would the optimal value of the objective function

change?

d$.$ If the available amount of labor decreased by $12$ hours, would that cause a change in the

optimal values of the decision variables or the optimal value of the objective function?

Would anything change?

e$.$ If the available amount of material increased by $10$ pounds to $210$ pounds, how would

that affect the optimal value of the objective function?

f$.$ If profit per unit on product $2$ increased by $$1$ and profit per unit on product $3$ decreased

by $$\mathrm{.}50,$ would that fall within the range of multiple changes?

g$.$ Would the values of the decision variables change?

h$.$ What would be the revised value of the objective function?