Chapter 12, Problem 22P

a)

Summary Introduction

To determine: The optimal order quantity with the minimum annual cost for purchasing the integrated chips.

Introduction:

Quantity discount model:

Quantity discount model is a scenario where manufacturing companies receive price discount for ordering large quantities. The firm will estimate if the discount quantity is profitable for the firm and make the decision.

Economic order quantity (EOQ):

EOQ is the quantity of units a company must add to its inventory so as to minimize the total inventory costs. It will determine the ideal order quantity which will decrease the inventory management costs.

Annual holding cost:

Annual holding cost is the cost involved in holding the excess inventory present within a firm. The excess inventory will be carried on for the next year and used for production.

Annual ordering cost:

Annual ordering cost is the cost involved in making a purchase order and the following up of that order. It will account for the labor involved and other associated costs.

Purchase cost:

Purchase cost is the cost incurred to purchase the require components to satisfy the demand of the firm. The cost will vary according to the demand.

Total cost:

Total cost is the overall cost taking into estimation of the holding, ordering and the purchase cost of the materials.

Answer to Problem 22P

The optimal order quantity is 200 units with the minimum annual cost of $1,446,380

Explanation of Solution

Given information:

Monthly demand (d)= 400 / month

Holding cost (H) = $35 / unit / year

Ordering cost (S) = $120 / order

Purchase price (P) = $350 / chip

$\begin{array}{c}\text{Annual demand (D)}=\text{Monthly demand}\times \text{12 months}\\ =400\times 12\\ =4,800\end{array}$

Price structure:

Quantity purchased	Price / unit
1 – 99 units	$350
100 – 199 units	$325
200 or more units	$300

Formula to calculate EOQ:

$\text{EOQ}=\sqrt{\frac{\text{2}\times \text{D}\times \text{S}}{\text{H}}}$

Formula to calculate annual holding cost:

$\text{Annual holding cost}=\frac{\text{EOQ}}{2}\times \text{H}$

Formula to calculate annual ordering cost:

$\text{Annual ordering cost}=\frac{\text{D}}{\text{EOQ}}\times \text{S}$

Formula to calculate purchase cost:

$\text{Purchase cost}=\text{D}\times \text{P}$

Formula to calculate total cost:

$\text{Total cost}=\text{Purchase cost}+\text{Annual ordering cost}+\text{Annual holding cost}$

Calculation of EOQ:

$\begin{array}{c}\text{EOQ}=\sqrt{\frac{2\times 4,800\times 120}{35}}\\ =\sqrt{\frac{1,152,000}{35}}\\ =\sqrt{32,914.285}\\ =181\text{ units}\end{array}$

The obtained EOQ value is in the range of (100 – 199 units). Hence, the price of that range will be used for calculation.

Calculation of values at (EOQ = 181):

Calculation of annual holding cost at (EOQ = 181):

$\begin{array}{c}\text{Annual holding cost}=\frac{181}{2}\times 35\\ =90.5\times 35\\ =3167.5\\ =\$3,168\end{array}$

Calculation of annual ordering cost at (EOQ = 181):

$\begin{array}{c}\text{Annual ordering cost}=\frac{4,800}{181}\times 120\\ =26.52\times 120\\ =\$3,182\end{array}$

Calculation of purchase cost:

$\begin{array}{c}\text{Purchase cost}=4,800\times 325\\ =\$1,560,000\end{array}$

Calculation of total cost at (EOQ = 181):

$\begin{array}{c}\text{Total cost}=1,560,000+3,168+3,182\\ =\$1,566,350\end{array}$

The total cost at the (EOQ = 181) is $1,566,350.

Calculation of values at discount quantity (200):

The next price break after the EOQ at 181 units is (200 or more units). The price of that range will be used in the following calculation.

Calculation of annual holding cost at (EOQ = 200):

$\begin{array}{c}\text{Annual holding cost}=\frac{200}{2}\times 35\\ =100\times 35\\ =\$3,500\end{array}$

Calculation of annual ordering cost at (EOQ = 200):

$\begin{array}{c}\text{Annual ordering cost}=\frac{4,800}{200}\times 120\\ =24\times 120\\ =\$2,880\end{array}$

Calculation of purchase cost:

$\begin{array}{c}\text{Purchase cost}=4,800\times 300\\ =\$1,440,000\end{array}$

Calculation of total cost at (EOQ = 200):

$\begin{array}{c}\text{Total cost}=1,440,000+3,500+2,880\\ =\$1,446,380\end{array}$

The total cost at the (EOQ = 200) is $1,446,380

Calculation of difference in cost:

The total cost at (EOQ = 181) is $1,566,350. The total cost at (EOQ = 200) is $1,446,380. Therefore (EOQ = 200) has lesser cost ($1,446,380 < 1,566,350).

Hence, the optimal order quantity is 200 units with the minimum annual cost of $1,446,380

b)

Summary Introduction

To determine: The optimal order quantity with the minimum annual cost for purchasing the integrated chips.

Answer to Problem 22P

The optimal order quantity is 200 units with the minimum annual cost of $1,445,880

Explanation of Solution

Given information:

Monthly demand (d)= 400 / month

Holding cost (H) = 10 % of purchase price of the price structure

Ordering cost (S) = $120 / order

Purchase price (P) = $350 / chip

$\begin{array}{c}\text{Annual demand (D)}=\text{Monthly demand}\times \text{12 months}\\ =400\times 12\\ =4,800\end{array}$

Price structure:

Quantity purchased	Price / unit
1 – 99 units	$350
100 – 199 units	$325
200 or more units	$300

Formula to calculate EOQ:

$\text{EOQ}=\sqrt{\frac{\text{2}\times \text{D}\times \text{S}}{\text{H}}}$

Formula to calculate annual holding cost:

$\text{Annual holding cost}=\frac{\text{EOQ}}{2}\times \text{H}$

Formula to calculate annual ordering cost:

$\text{Annual ordering cost}=\frac{\text{D}}{\text{EOQ}}\times \text{S}$

Formula to calculate purchase cost:

$\text{Purchase cost}=\text{D}\times \text{P}$

Formula to calculate total cost:

$\text{Total cost}=\text{Purchase cost}+\text{Annual ordering cost}+\text{Annual holding cost}$

Calculation of EOQ at each price structure:

EOQ at price = $350

$\begin{array}{c}{\text{EOQ}}_1=\sqrt{\frac{2\times 4,800\times 120}{\left(10\%\times 350\right)}}\\ =\sqrt{\frac{1,152,000}{35}}\\ =\sqrt{32,914.285}\\ =181\text{ units}\end{array}$

EOQ at price = $325

$\begin{array}{c}{\text{EOQ}}_2=\sqrt{\frac{2\times 4,800\times 120}{\left(10\%\times 325\right)}}\\ =\sqrt{\frac{1,152,000}{32.5}}\\ =\sqrt{35,446.15}\\ =188\text{ units}\end{array}$

EOQ at price = $300

$\begin{array}{c}{\text{EOQ}}_3=\sqrt{\frac{2\times 4,800\times 120}{\left(10\%\times 300\right)}}\\ =\sqrt{\frac{1,152,000}{30}}\\ =\sqrt{38,400}\\ =196\text{ units}\end{array}$

EOQ = 181, cannot be bought at the price of $350. EOQ = 196 cannot be bought at the price of $300.

Hence, the EOQ is 188 units.

The obtained EOQ value is in the range of (100 – 199 units). Hence, the price of that range will be used for calculation.

Calculation of values at (EOQ = 188):

Calculation of annual holding cost at (EOQ = 188):

$\begin{array}{c}\text{Annual holding cost}=\frac{188}{2}\times \left(10\%\times 325\right)\\ =94\times 32.5\\ =\$3,055\end{array}$

Calculation of annual ordering cost at (EOQ = 188):

$\begin{array}{c}\text{Annual ordering cost}=\frac{4,800}{188}\times 120\\ =25.53\times 120\\ =\$3,064\end{array}$

Calculation of purchase cost:

$\begin{array}{c}\text{Purchase cost}=4,800\times 325\\ =\$1,560,000\end{array}$

Calculation of total cost at (EOQ = 188):

$\begin{array}{c}\text{Total cost}=1,560,000+3,055+3,064\\ =\$1,566,119\end{array}$

The total cost at the (EOQ = 188) is $1,566,319.

Calculation of values at discount quantity (200):

The next price break after the EOQ at 188 units is (200 or more units). The price of that range will be used in the following calculation.

Calculation of annual holding cost at (EOQ = 200):

$\begin{array}{c}\text{Annual holding cost}=\frac{200}{2}\times 30\\ =100\times 30\\ =\$3,000\end{array}$

Calculation of annual ordering cost at (EOQ = 200):

$\begin{array}{c}\text{Annual ordering cost}=\frac{4,800}{200}\times 120\\ =24\times 120\\ =\$2,880\end{array}$

Calculation of purchase cost:

$\begin{array}{c}\text{Purchase cost}=4,800\times 300\\ =\$1,440,000\end{array}$

Calculation of total cost at (EOQ = 200):

$\begin{array}{c}\text{Total cost}=1,440,000+3,000+2,880\\ =\$1,445,880\end{array}$

The total cost at the (EOQ = 200) is $1,445,880

Calculation of difference in cost:

The total cost at (EOQ = 188) is $1,566,319. The total cost at (EOQ = 200) is $1,445,880. Therefore (EOQ = 200) has lesser cost ($1,445,880 < 1,566,319).

Hence, the optimal order quantity is 200 units with the minimum annual cost of $1,445,880.