Chapter 12, Problem 25P

a)

Summary Introduction

To determine: The number of tires Company RMT should order each time it places an order.

Introduction: Inventory management is the process of ordering, storing and using inventory of the company such raw material, components and finished goods. It governs the flow of goods from manufacturers to warehouse and to the point of sale. The key function is to maintain record of flow of new or returned products which enters or leaves the company.

Answer to Problem 25P

The number of tires Company RMT should order each time it places an order is 667.

Explanation of Solution

Given information:

$\begin{array}{c}\text{Annual}\text{demand,D}=\text{20,000}\\ \text{Order}\text{cost,S}=\text{\$40}\text{per}\text{order}\end{array}$

$\begin{array}{c}\text{Price}\text{for}\text{order}\text{less}\text{than}\text{500 tires}=\text{\$20/tire}\\ \text{Price}\text{for}\text{order}\text{between}\text{500}\text{to}\text{999}\text{tires}=\text{\$18/tire}\\ \text{Price}\text{for}\text{order}\text{more than}\text{1,000}\text{tires}=\text{\$17/tire}\end{array}$

Formula:

$\text{EOQ,Q}=\sqrt{\frac{\text{2DS}}{\text{H}}}$

Where

$\begin{array}{c}\text{D}=\text{Demand}\\ \text{S}=\text{Ordering}\text{cost}\\ \text{H}=\text{Holding}\text{cost}\end{array}$

Calculation of optimal order quantity:

Calculation of EOQ for orders less than 500 tires:

$\begin{array}{c}{\text{Q}}_{\text{1}}=\sqrt{\frac{\text{2DS}}{\text{H}}}\\ =\sqrt{\frac{2\times 20,000\times 40}{0.2\times 20}}\\ =632.5\text{tires}\end{array}$

EOQ for order less than 500 tires is calculated by multiplying 2, 20,000 and 40 and dividing the resultant with the product of 0.2 and 20 and taking square root which gives 632.5 tires.

Calculation of EOQ for orders between 500 to 999 tires:

$\begin{array}{c}{\text{Q}}_2=\sqrt{\frac{\text{2DS}}{\text{H}}}\\ =\sqrt{\frac{2\times 20,000\times 40}{0.2\times 18}}\\ =666.7\text{tires}\\ \text{=667}\text{tires }\left(\text{approx}\right)\end{array}$

EOQ for order between 500 to 999 tires is calculated by multiplying 2, 20,000 and 40 and dividing the resultant with the product of 0.2 and 18 and taking square root which gives 667 tires.

Calculation of EOQ for orders more than 1,000 tires:

$\begin{array}{c}{\text{Q}}_3=\sqrt{\frac{\text{2DS}}{\text{H}}}\\ =\sqrt{\frac{2\times 20,000\times 40}{0.2\times 17}}\\ =686\text{tires}\end{array}$

EOQ for order orders more than 1,000 tires is calculated by multiplying 2, 20,000 and 40 and dividing the resultant with the product of 0.2 and 17 and taking square root which gives 686 tires.

Based on the above calculations, EOQ of orders between 500 to 999 tires gives optimal order quantity as the EOQ lies within the quantities ordered. Therefore the optimal order quantity is 667 tires at $0.18.

Hence, the number of tires Company RMT should order each time it places an order is 667.

b)

Summary Introduction

To determine: The total annual cost of the policy.

Answer to Problem 25P

The total annual cost of the policy is $342,500 per year.

Explanation of Solution

Given information:

$\begin{array}{c}\text{Annual}\text{demand}=\text{45,000}\text{pieces}\\ \text{Order}\text{cost}=\text{\$200}\text{per}\text{order}\end{array}$

$\begin{array}{c}\text{Price}\text{for}\text{order}\text{less}\text{than}\text{2000}\text{pieces}=\text{\$7}\text{.80/piece}\\ \text{Price}\text{for}\text{order}\text{between2,001}\text{to}\text{5000}\text{pieces}=\text{\$1}\text{.60/piece}\\ \text{Price}\text{for}\text{order}\text{between5,001}\text{to}\text{10,000}\text{pieces}=\text{\$1}\text{.40/piece}\\ \text{Price}\text{for}\text{order}\text{more than}\text{10,001}\text{pieces}=\text{1}\text{.25/piece}\end{array}$

Formula:

$\text{Total}\text{cost}=\text{PD}+\frac{\text{Q}}{\text{2}}\times \text{H+}\frac{\text{D}}{\text{Q}}\times \text{S}$

Calculation of total annual cost:

The total annual cost for ordering quantity 667 tires is compared with 1000 tires to find the optimal total cost and quantities to be ordered by the company.

Total annual cost for 667 tires:

$\begin{array}{c}\text{Total}\text{cost}=\left(\$18\times 20,000\right)+\left(\frac{667}{2}\times \left(0.2\times 18\right)\right)+\left(\frac{20,000}{667}\times \$40\right)\\ =\$360,000+\$1,200+\$1,200\\ =\$362,400/\text{year}\end{array}$

Total annual cost is calculated by adding holding cost, ordering cost and unit cost and the values are substituted in the above formula which yields $362,400 as total cost per year.

The total annual cost for 667 ordering quantity is $362,400/year.

Total annual cost for 1000 tires:

$\begin{array}{c}\text{Total}\text{cost}=\left(\$17\times 20,000\right)+\left(\frac{1,000}{2}\times \left(0.2\times 17\right)\right)+\left(\frac{20,000}{1,000}\times \$40\right)\\ =\$340,000+\$1,700+\$800\\ =\$342,500/\text{year}\end{array}$

Total annual cost is calculated by adding holding cost, ordering cost and unit cost and the values are substituted in the above formula which yields $342,500 as total cost per year.

The total annual cost for 1000 tires is $342,500/year.

The total annual cost for 1000 tires is lesser than total annual cost for 667 tires and so company RMT should order 1,000 tires each time.

Hence, the total annual cost of the policy is $342,500 per year.