Let's walk through the steps for calculating the optimal order quantity, price per unit, and total cost $($including ordering and holding inventory$)$ given the information you provided.

Given Information:

Demand $($D$)=12,550$ units

Order Cost $($S$)=$ $$175$ per order

Holding Cost per unit $=8\%$ of the unit price

Price Ranges:

$1-999$ units: Price $=$ $$91$ per unit$1000-3499$ units: Price $=$ $$88$ per unit$3500-5999$ units: Price $=$ $$86$ per unit

Step $1$: Calculate Holding Cost $($H$)$ for each price range

The holding cost per unit is $8\%$ of the unit price.

For $1-999$ units $($Price $=$ $$91)$:

H$=0\mathrm{.}0891=7\mathrm{.}28$per$$unit$$per$$yearH $=0\mathrm{.}08\backslash$times $91=7\mathrm{.}28\backslash ,\backslash$text$\{$per unit per year$\}$H$=0\mathrm{.}0891=7\mathrm{.}28$per$$unit$$per$$year

For $1000-3499$ units $($Price $=$ $$88)$:

H$=0\mathrm{.}0888=7\mathrm{.}04$per$$unit$$per$$yearH $=0\mathrm{.}08\backslash$times $88=7\mathrm{.}04\backslash ,\backslash$text$\{$per unit per year$\}$H$=0\mathrm{.}0888=7\mathrm{.}04$per$$unit$$per$$year

For $3500-5999$ units $($Price $=$ $$86)$:

H$=0\mathrm{.}0886=6\mathrm{.}88$per$$unit$$per$$yearH $=0\mathrm{.}08\backslash$times $86=6\mathrm{.}88\backslash ,\backslash$text$\{$per unit per year$\}$H$=0\mathrm{.}0886=6\mathrm{.}88$per$$unit$$per$$year

Step $2$: Calculate the Economic Order Quantity $($EOQ$)$

The EOQ formula helps us determine the optimal quantity to order in order to minimize the total cost of inventory $($ordering and holding costs$).$

EOQ$=2$DSHEOQ $=\backslash$sqrt$\{\backslash$frac$\{2$DS$\}\{$H$\}\}$EOQ$=$H$2$DS$$

Where:

D $=12,550($demand$)$

S $=175($ordering cost$)$

H $=$ Holding cost per unit $($calculated above$)$

For Price Range $1-999$ units $($Price $=$ $$91,$ Holding Cost $=$ $$7\mathrm{.}28)$:

EOQ$=212,5501757\mathrm{.}28=4,387,5007\mathrm{.}28602,325\mathrm{.}27776\mathrm{.}6$unitsEOQ $=\backslash$sqrt$\{\backslash$frac$\{2\backslash$times $12,550\backslash$times $175\}\{7\mathrm{.}28\}\}=\backslash$sqrt$\{\backslash$frac$\{4,387,500\}\{7\mathrm{.}28\}\}\backslash$approx $\backslash$sqrt$\{602,325\mathrm{.}27\}\backslash$approx $776\mathrm{.}6\backslash ,\backslash$text$\{$units$\}$EOQ$=7\mathrm{.}28212,550175=7\mathrm{.}284,387,500602,325\mathrm{.}27776\mathrm{.}6$units

So$,$ the optimal order quantity is $777$ units.

For Price Range $1000-3499$ units $($Price $=$ $$88,$ Holding Cost $=$ $$7\mathrm{.}04)$:

EOQ$=212,5501757\mathrm{.}04=4,387,5007\mathrm{.}04623,442\mathrm{.}05790\mathrm{.}8$unitsEOQ $=\backslash$sqrt$\{\backslash$frac$\{2\backslash$times $12,550\backslash$times $175\}\{7\mathrm{.}04\}\}=\backslash$sqrt$\{\backslash$frac$\{4,387,500\}\{7\mathrm{.}04\}\}\backslash$approx $\backslash$sqrt$\{623,442\mathrm{.}05\}\backslash$approx $790\mathrm{.}8\backslash ,\backslash$text$\{$units$\}$EOQ$=7\mathrm{.}04212,550175=7\mathrm{.}044,387,500623,442\mathrm{.}05790\mathrm{.}8$units

So$,$ the optimal order quantity is $1000$ units.

For Price Range $3500-5999$ units $($Price $=$ $$86,$ Holding Cost $=$ $$6\mathrm{.}88)$:

EOQ$=212,5501756\mathrm{.}88=4,387,5006\mathrm{.}88638,365\mathrm{.}22798\mathrm{.}98$unitsEOQ $=\backslash$sqrt$\{\backslash$frac$\{2\backslash$times $12,550\backslash$times $175\}\{6\mathrm{.}88\}\}=\backslash$sqrt$\{\backslash$frac$\{4,387,500\}\{6\mathrm{.}88\}\}\backslash$approx $\backslash$sqrt$\{638,365\mathrm{.}22\}\backslash$approx $798\mathrm{.}98\backslash ,\backslash$text$\{$units$\}$EOQ$=6\mathrm{.}88212,550175=6\mathrm{.}884,387,500638,365\mathrm{.}22798\mathrm{.}98$units

So$,$ the optimal order quantity is $3500$ units.

Step $3$: Calculate Total Cost $($TC$)$

The total cost $($TC$)$ is the sum of the ordering cost and the holding cost:

TC$=($DQ$)$S$+($Q$2)$HTC $=\backslash$left$(\backslash$frac$\{$D$\}\{$Q$\}\backslash$right$)\backslash$times S $+\backslash$left$(\backslash$frac$\{$Q$\}\{2\}\backslash$right$)\backslash$times HTC$=($QD$)$S$+(2$Q$)$H

Where:

D $=12,550$ units $($demand$)$

S $=175($ordering cost$)$

H $=$ Holding cost per unit $($calculated above$)$

Q $=$ Optimal order quantity

For $1-999$ units $($Price $=$ $$91,$ Holding Cost $=$ $$7\mathrm{.}28,$ Q $=777$ units$)$:

Ordering cost:

$12,550777=16\mathrm{.}15($orders$$per$$year$)\backslash$frac$\{12,550\}\{777\}=16\mathrm{.}15\backslash$quad $\backslash$text$\{($orders per year$)\}77712,550=16\mathrm{.}15($orders$$per$$year$)16\mathrm{.}15175=2,826\mathrm{.}2516\mathrm{.}15\backslash$times $175=2,826\mathrm{.}2516\mathrm{.}15175=2,826\mathrm{.}25$

Holding cost:

$77727\mathrm{.}28=2,828\mathrm{.}52\backslash$frac$\{777\}\{2\}\backslash$times $7\mathrm{.}28=2,828\mathrm{.}5227777\mathrm{.}28=2,828\mathrm{.}52$

Total cost:

TC$=2,826\mathrm{.}25+2,828\mathrm{.}52=5,654\mathrm{.}77$TC $=2,826\mathrm{.}25+2,828\mathrm{.}52=5,654\mathrm{.}77$TC$=2,826\mathrm{.}25+2,828\mathrm{.}52=5,654\mathrm{.}77$

For $1000-3499$ units