T$.$J$.$ Maxx, a discount retail chain, is planning to construct a new store in South Bend. The land they've acquired for this purpose is spacious enough to house a store with a total floor area of $140,000$ square feet. Using insights from marketing and demographic studies conducted in the region, along with historical performance data from their existing stores, T$.$J$.$ Maxx has calculated the expected annual profit per square foot for each department within the store, as detailed in the table below.

Department Profit per square feet $($$$)$

Men's clothing

$($x$1)$

$4\mathrm{.}25$

Women's clothing

$($x$2)$

$5\mathrm{.}10$

Children's clothing

$($x$3)$

$4\mathrm{.}50$

Toys

$($x$4)$

$5\mathrm{.}20$

Housewares

$($x$5)$

$4\mathrm{.}10$

Electronics

$($x$6)$

$4\mathrm{.}90$

Auto supplies

$($x$7)$

$3\mathrm{.}80$

Each department must have at least $15,000$ square feet of floor space, and no department can have more than $20\%$ of the total retail floor space. Men's, women's, and children's clothing plus housewares keep all their stock on the retail floor; however, toys, electronics, and auto supplies keep some items $($such as bicycles, televisions, and tires$)$ in inventory. Thus, $10\%$ of the total retail floor space devoted to these three departments must be set aside outside the retail area for stocking inventory. T$.$J$.$ Maxx wants to know the floor space that should be devoted to each department to maximize profit. Let

x$8$

be the floor space devoted for stocking inventory.

The linear programming model for the problem is as follows:

max $4\mathrm{.}25$x$1+5\mathrm{.}10$x$2+4\mathrm{.}50$x$3+5\mathrm{.}20$x$4+4\mathrm{.}10$x$5+4\mathrm{.}90$x$6+3\mathrm{.}80$x$7$

s$.$t$.$

$8$

xi

i$=1$

$<=140,000$

xi $>=15,000$; i $=1,2,,7$

xi

$7$

xi

i$=1$

$<=0\mathrm{.}20$; i $=1,2,,7$

x$8$

x$4+$ x$6+$ x$7$

$=0\mathrm{.}10$

xi $>=0$; i $=1,2,,7$

Use the attached R output to answer the questions that follow.

What are the optimal floor spaces $($in square feet$)?($Enter the values exactly as shown in the output.$)$