Chapter 12, Problem 3PA

Joe Birra needs to purchase malt for his microbrewery production. His supplier charges $35 per delivery (no matter how much is delivered) and $1.20 per gallon. Joe’s annual holding cost per unit is 35 percent of the price per gallon. Joe uses 250 gallons of malt per week.

1. a. Suppose Joe orders 1000 gallons each time. What is his average inventory (in gal)?
2. b. Suppose Joe orders 1500 gallons each time. How many orders does he place with his supplier each year?
3. c. How many gallons should Joe order from his supplier with each order to minimize the sum of the ordering and holding costs?
4. d. Suppose Joe orders 2500 gallons each time he places an order with the supplier. What is the sum of the ordering and holding costs per gallon?
5. e. Suppose Joe orders the quantity from part (C) that minimizes the sum of the ordering and holding costs each time he places an order with the supplier. What is the annual cost of the EOQ expressed as a percentage of the annual purchase cost?
6. f. If Joe’s supplier only accepts orders that are an integer multiple of 1000 gallons, how much should Joe order to minimize ordering and holding costs per gallon?
7. g. Joe’s supplier offers a 3 percent discount if Joe is willing to purchase 8000 gallons or more. What would Joe’s total annual cost (purchasing, ordering, and holding) be if he were to take advantage of the discount?

Summary Introduction

To determine: The average inventory in gallon.

Explanation of Solution

Given information:

Weekly demand (d)         = 250 gallons

Purchase price (P) per gallon    = $1.20

Order cost (O) per order    = $35

Annual holding cost (H)     = 35%

Weeks per year (W)         = 52

Order quantity (Q)         = 1,000 gallons

Calculation of average inventory:

$\begin{array}{c}\text{Average inventory}=\frac{Q}{2}\\ =\frac{1,000}{2}\\ =500\text{ gallons}\end{array}$

The average inventory is 500 gallons.

Summary Introduction

To determine: The number of orders placed each year.

Explanation of Solution

Given information:

Weekly demand (d)         = 250 gallons

Purchase price (P) per gallon    = $1.20

Order cost (O) per order    = $35

Annual holding cost (H)     = 35%

Weeks per year (W)         = 52

Order quantity (Q)         = 1,500 gallons

Calculation of number of orders each year:

$\begin{array}{c}\text{Number of orders each year}=\frac{d\times W}{Q}\\ =\frac{250\times 52}{1,500}\\ =8.67\text{ orders}\end{array}$

The number of orders each year is 8.67 orders.

Summary Introduction

To determine: The gallons that must be ordered in each order to minimize the sum of ordering and holding costs.

Explanation of Solution

Given information:

Weekly demand (d)         = 250 gallons

Purchase price (P) per gallon    = $1.20

Order cost (O) per order    = $35

Annual holding cost (H)     = 35%

Weeks per year (W)         = 52

Calculation of order quantity:

$\begin{array}{c}\text{Order quantity}=\sqrt{\frac{2\times \left(d\times W\right)\times O}{H\times P}}\\ =\sqrt{\frac{2\times \left(250\times 52\right)\times 35}{\frac{35}{100}\times 1.2}}\\ =\sqrt{\frac{2\times 13,000\times 35}{0.42}}\\ =\sqrt{2,166,666.67}\\ =1,471.9\\ =1,472\text{ gallons}\end{array}$

The gallons that must be ordered in each order to minimize the sum of ordering and holding costs is 1,472 gallons.

Summary Introduction

To determine: The sum of the annual ordering cost and holding costs per gallon.

Explanation of Solution

Given information:

Weekly demand (d)         = 250 gallons

Purchase price (P) per gallon    = $1.20

Order cost (O) per order    = $35

Annual holding cost (H)     = 35%

Weeks per year (W)         = 52

Order quantity (Q)         = 2,500 gallons

Calculation of sum of annual ordering cost and holding costs:

Annual ordering cost:

$\begin{array}{c}\text{Cost}=\frac{d\times W}{Q}\times O\\ =\frac{250\times 52}{2,500}\times 35\\ =\frac{13,000}{2,500}\times 35\\ =\$182\end{array}$

Annual holding cost:

$\begin{array}{c}\text{Cost}=\frac{Q}{2}\times \left(H\times P\right)\\ =\frac{2,500}{2}\times \left(\frac{35}{100}\times 1.2\right)\\ =\frac{2,500}{2}\times 0.42\\ =\$525\end{array}$

Sum of annual ordering cost and holding cost:

$\begin{array}{c}\text{Sum}=\frac{\$182+\$525}{d\times W}\\ =\frac{\$707}{250\times 52}\\ =\frac{\$707}{13,000}\\ =\$0.0544\text{ per gallon}\end{array}$

The sum of the annual ordering cost and holding costs per gallon is $0.0544.

Summary Introduction

To determine: The annual cost of EOQ expressed as a percentage of the annual purchase cost.

Explanation of Solution

Given information:

Weekly demand (d)         = 250 gallons

Purchase price (P) per gallon    = $1.20

Order cost (O) per order    = $35

Annual holding cost (H)     = 35%

Weeks per year (W)         = 52

Order quantity (Q)         = 1,472 gallons

Calculation of sum of annual ordering cost and holding costs:

Annual ordering cost:

$\begin{array}{c}\text{Cost}=\frac{d\times W}{Q}\times O\\ =\frac{250\times 52}{1,472}\times 35\\ =\frac{13,000}{1,472}\times 35\\ =\$309.10\end{array}$

Annual holding cost:

$\begin{array}{c}\text{Cost}=\frac{Q}{2}\times \left(H\times P\right)\\ =\frac{1,472}{2}\times \left(\frac{35}{100}\times 1.2\right)\\ =736\times 0.42\\ =\$309.12\end{array}$

Annual purchase cost:

$\begin{array}{c}\text{Cost}=P\times d\times W\\ =1.20\times 250\times 52\\ =\$15,600\end{array}$

Annual cost of EOQ as a percentage of the annual purchase cost:

$\begin{array}{c}\text{Percentage}=\frac{\text{Annual ordering cost}+\text{Annual holding cost}}{\text{Purhcase cost}}\times 100\\ =\frac{309.10+309.12}{15,600}\times 100\\ =\frac{618.22}{15,600}\times 100\\ =3.96\%\end{array}$

The annual cost of EOQ expressed as a percentage of the annual purchase cost is 3.96%.

Summary Introduction

To determine: The ordering quantity if the supplier only accepts orders in the integer multiple of 1,000 gallons.

Explanation of Solution

Given information:

Weekly demand (d)         = 250 gallons

Purchase price (P) per gallon    = $1.20

Order cost (O) per order    = $35

Annual holding cost (H)     = 35%

Weeks per year (W)         = 52

Calculation of sum of annual ordering cost and holding costs:

The EOQ is 1,472 gallons. The cost is calculated for 1,000 gallons and 2,000 gallons and the quantity with the lowest cost is selected.

Cost when Q is 1,000 gallons:

Annual ordering cost:

$\begin{array}{c}\text{Cost}=\frac{d\times W}{Q}\times O\\ =\frac{250\times 52}{1,000}\times 35\\ =\frac{13,000}{1,000}\times 35\\ =\$455\end{array}$

Annual holding cost:

$\begin{array}{c}\text{Cost}=\frac{Q}{2}\times \left(H\times P\right)\\ =\frac{1,000}{2}\times \left(\frac{35}{100}\times 1.2\right)\\ =500\times 0.42\\ =\$210\end{array}$

Sum of annual ordering cost and holding cost:

$\begin{array}{c}\text{Sum}=\$455+\$210\\ =\$665\end{array}$

Cost when Q is 2,000 gallons:

Annual ordering cost:

$\begin{array}{c}\text{Cost}=\frac{d\times W}{Q}\times O\\ =\frac{250\times 52}{1,000}\times 35\\ =\frac{13,000}{2,000}\times 35\\ =\$227.50\end{array}$

Annual holding cost:

$\begin{array}{c}\text{Cost}=\frac{Q}{2}\times \left(H\times P\right)\\ =\frac{2,000}{2}\times \left(\frac{35}{100}\times 1.2\right)\\ =1,000\times 0.42\\ =\$420\end{array}$

Sum of annual ordering cost and holding cost:

$\begin{array}{c}\text{Sum}=\$227.50+\$520\\ =\$647.50\end{array}$

An order size of 2,000 gallons has lower cost compared to an order size of 1,000 gallons ($647.50 < $665). Therefore, the best order quantity is 1,000 units.

The ordering quantity if the supplier only accepts orders in the integer multiple of 1,000 gallons is 1,000 gallons.

Summary Introduction

To determine: The sum of the annual ordering cost, purchase cost, and holding costs if Person J takes advantage of the discount.

Explanation of Solution

Given information:

Weekly demand (d)         = 250 gallons

Purchase price (P) per gallon    = $1.20

Order cost (O) per order    = $35

Annual holding cost (H)     = 35%

Weeks per year (W)         = 52

Discount % (D)         = 3%

Order quantity (Q)         = 8,000 gallons

Calculation of sum of annual ordering cost, purchase cost, and holding costs:

Annual ordering cost:

$\begin{array}{c}\text{Cost}=\frac{d\times W}{Q}\times O\\ =\frac{250\times 52}{8,000}\times 35\\ =\frac{13,000}{8,000}\times 35\\ =\$56.88\end{array}$

Annual holding cost:

$\begin{array}{c}\text{Cost}=\frac{Q}{2}\times \left[H\times \left(P-\left(D\times P\right)\right)\right]\\ =\frac{8,000}{2}\times \left[\frac{35}{100}\times \left(1.20-\left(\frac{3}{100}\times 1.20\right)\right)\right]\\ =\frac{8,000}{2}\times \left[\frac{35}{100}\times \left(1.20-0.036\right)\right]\\ =\frac{8,000}{2}\times \left(\frac{35}{100}\times 1.164\right)\\ =\frac{8,000}{2}\times 0.4074\\ =\$1,629.60\end{array}$

Purchase cost

$\begin{array}{c}\text{Cost}=\left(P-\left(P\times D\right)\right)\times d\times W\\ =\left(1.20-\left(1.20\times \frac{3}{100}\right)\right)\times 250\times 52\\ =\left(1.20-0.036\right)\times 250\times 52\\ =1.164\times 250\times 52\\ =\$15,132\end{array}$

Sum of annual ordering cost, purchase cost, and holding cost:

$\begin{array}{c}\text{Sum}=\$56.88+\$1,629.60+\$15,132\\ =\$16,818.48\end{array}$

The sum of the annual ordering cost, purchase cost, and holding costs is $16,818.48.