: A small mail$-$order company uses $20000$ boxes a year $(250$ working days for a year$).$ Holding cost rate is $20\%$ of unit price per year, and ordering cost is $$16$ per order. The lead time

is $2$ days. The following quantity discount is available. Determine:

Number of Boxes $=$ Price per Box

Less than $2000=$ $$1\mathrm{.}8$

$2000$ to $4999=$ $$1\mathrm{.}6$

$5000$ or more $=$ $$1\mathrm{.}4$

What is the optimal ordering policy.

Where:

D $=$ Demand in units per year $(20,000)$

S $=$ Ordering cost per order $(16)$

H $=$ Holding cost rate $(20$

Explanation:

First, let's calculate the unit price:

For less than $2000$ boxes:$1\mathrm{.}8$

For $2000$ to $4999$ boxes: $1\mathrm{.}6$

For $5000$ or more boxes:$1\mathrm{.}4$

Next, we need to calculate the EOQ for each price level and compare the total cost for each to find the optimal ordering policy.

For less than $2000$ boxes:

EOQ $=((2*20000*16)/0\mathrm{.}20*1\mathrm{.}8)=1825$ units

For $2000$ to $4999$ boxes:

EOQ $=((2*20000*16)/0\mathrm{.}20*1\mathrm{.}6)=2025$ units

For $5000$ or more boxes:

EOQ $=((2*20000*16)/0\mathrm{.}20*1\mathrm{.}4)=2182$ units

Step $2$

Now, let's calculate the total cost for each EOQ:

Total cost $=($D$/$Q$)*$S $+($Q$/2)*$H $+($D$/$Q$)*$P

Where:

Q $=$ EOQ

P $=$ Unit price

Explanation:

For less than $2000$ boxes:

Total cost $=(20000/1825)*16+(1825/2)*(0\mathrm{.}20*1\mathrm{.}8)+(20000/1825)*1\mathrm{.}8=3,510$

For $2000$ to $4999$ boxes:

Total cost $=(20000/2025)*16+(2025/2)*(0\mathrm{.}20*1\mathrm{.}6)+(20000/2025)*1\mathrm{.}6=3,480$

For $5000$ or more boxes:

Total cost $=(20000/2182)*16+(2182/2)*(0\mathrm{.}20*1\mathrm{.}4)+(20000/2182)*1\mathrm{.}4=3,473$

Answer

Based on the calculations, the optimal ordering policy is to order $5000$ or more boxes at a time, with a reorder point of $2182$ units. This will result in the lowest total cost for the company.