Problem $1(20$ points$)$

Each year the Smalltown branch $of$ a furniture store sells $30,000$ dining room tables. The local store orders tables from the company warehouse providing stock for many local branches and the cost $of$ doing this $is\frac{?}{(\frac{?}{?}\text{\$}15\frac{?}{?})}$ per table plus a fixed cost $of\frac{?}{(\frac{?}{?}\text{\$}100\frac{?}{?})}$ per order. Assume that the lead time $is$ zero. The store's manager believes that the demand for tables can $be$ backlogged and that the cost $of$ being short one table for one year $is\frac{?}{(\frac{?}{?}\text{\$}25\frac{?}{?}).}$ The annual holding cost for inventory $is30c$ per dollar value $of$ inventory.

$(a)$ What $is$ the optimal order quantity?

$(b)If$ the store manager orders the order quantity found $in$ part $(a),$ what $is$ the maximum shortage that will occur?

$(c)At$ most, how many tables can the Smalltown furniture store have $in$ stock $if$ the optimal ordering policy $is$ followed?

$(d)$ What $is$ the average cost per year under the optimal policy?

Problem $2(15$ points$)$

Three years ago, Panera Bread opened a franchise $on$ Plymouth Road. They bake their bread fresh every morning very early before the store opens, and they don't make more throughout the day. A loaf costs $\frac{?}{(\frac{?}{?}\text{\$}2\frac{?}{?})}to$ bake and can $be$ sold $to$ a customer for $\frac{?}{(\frac{?}{?}\text{\$}5\frac{?}{?}).}$ Any loaves that are leftover $at$ the end $of$ the day can $be$ donated $to$ a local food bank and will have $a\frac{?}{(\frac{?}{?}\text{\$}0\mathrm{.}50\frac{?}{?})}$ tax benefit. The manager, $Ed,$ has been estimating how much $to$ bake each day based $on$ his expert judgment. However, the store has also been recording how many loaves $of$ bread are ordered over the past several months, and $he$ would like $to$ start using this data $to$ determine how much $to$ bake.

When $Ed$ looks $at$ the data, $he$ notices that the number $of$ loaves ordered $on$ Tuesdays $is$ uniformly distributed between $350$ and $420$ loaves. How many loaves should this Panera location bake $on$ Tuesday mornings? Hint: the quantile function $ofan\frac{?}{(U(a,b)\frac{?}{?})}is\frac{?}{(F^{-1}(t))}=a+t(b-a)\frac{?}{?}.$

Problem $3(30$ points$)$

Formulate the following problem $as$ a linear program and solve using the Excel Solver. Repport the

optimal solution and profit.

You're working $asan$ Industrial Engineer $at$ Dell. Customers purchase laptops throughout the year,

but before each semester, university bookstores order large quantities $to$ have $in$ stock for students

and staff. The business intelligence unit just released the initial demand forecasts for January orders

$in$ the Midwest region, and your manager, Alejandro, stopped $by$ your office $to$ discuss the production

plans. He'd like $to$ know the most profitable way $to$ fill all $of$ the orders and asked you $to$ send him $a$

report $by$ next Thursday. $He$ wants a breakdown $of$ how many units $of$ each type $of$ laptop the

manufacturing division should plan $to$ produce. Because it's a draft plan, Alejandro $is$ fine with non$-$

integer estimates.

They have the option $to$ produce one $of$ four types $in$ the $XPS$ product line. There are two sizes: $13in$

and $15in,$ and both have a touchscreen and non$-$touchscreen option. The prices $of$ the non$-$

touchscreen laptops are $$800$ and $$900$ for $13in$ and $15in,$ respectively, and the price $of$ the

touchscreen option $on$ either size $isan$ additional $$200.$ The production costs $of$ the $13in$ laptop are

$$600(non-$touchscreen$)$ and $$650(touch),$ and the $15in$ laptop costs $$625(non-$touch$)$ and $$675$

$(touch).$

The forecast predicts a total $of10,000$ units $tobe$ ordered. $It$ says $4,000$ will $be$ for $13in$ laptops,

$2,000$ will $be$ for $15in$ laptops, and the remaining $4,000$ units won't have a size specified. They predict

exactly $3,000$ touchscreens will $be$ ordered but can't say what size. Space may also $be$ a $limiting$

factor $-$ Dell has set aside $3,800$ cubic feet $of$ storage space for these orders. The boxes for the $13in$

laptops take $up0\mathrm{.}3$ cubic feet, and the boxes for the $15in$ laptops take $up0\mathrm{.}45$ cubic feet.

Problem $4(30$ points$)$

Consider the following linear programming formulation:

min,$x+4y$

$s.t2y+x44y-$

$x-1x5x$

$0y0$

$(a)$ Graph all constraints and clearly identify the feasible region for the solution. Make sure $to$

clearly label which line $on$ the graph corresponds $to$ each constraint.

$(b)$ Solve the $LP$ using the graphical method $(also$ called the iso$-$profit line method$)$ and the corner

$(extreme)$ point method and state the optimal solution and the optimal objective function

value.

$(c)$ Which constraints are binding $in$ the optimal solution?

$(d)$ Consider removing the $y0$ constraint from the problem. Does doing this change the feasible

region? Explain why $or$ why not.