Chapter 12, Problem 49P

A gourmet coffee shop in downtown San Francisco is open 200 days a year and sells an average of 75 pounds of Kona coffee beans a day. (Demand can be assumed to be distributed normally, with a standard deviation of 15 pounds per day.) After ordering (fixed cost = $16 per order), beans are always shipped from Hawaii within exactly 4 days. Per-pound annual holding costs for the beans are $3.

a) What is the economic order quantity (EOQ) for Kona coffee beans?

b) What are the total annual holding costs of stock for Kona coffee beans?

c) What are the total annual ordering costs for Kona coffee beans?

d) Assume that management has specified that no more than a 1% risk during stockout is acceptable. What should the reorder point (ROP) be?

e) What is the safety stock needed to attain a 1% risk of stockout during lead time?

f) What is the annual holding cost of maintaining the level of safety stock needed to support a 1% risk?

g) If management specified that a 2% risk of stockout during lead time would be acceptable, would the safety stock holding costs decrease or increase?

Summary Introduction

To determine: The Economic order quantity.

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 49P

The Economic order quantity is 400 lb. of beans.

Explanation of Solution

Given information:

$\begin{array}{c}\text{Annual}\text{demand,D}=75\times 200\\ =15,000\text{lb/year}\\ \text{Setup}\text{cost,S}=\$16\text{/order}\\ \text{Holding}\text{cost,H}=\$3\text{lb/year}\end{array}$

$\begin{array}{c}\text{Standard}\text{deviation,}\text{σ}=\text{15}\text{pounds/day}\\ \text{Lead}\text{time, L}=\text{4}\text{days}\\ \text{Number}\text{of}\text{working}\text{days}=\text{200}\end{array}$

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:

$\begin{array}{c}\text{Q}=\sqrt{\frac{\text{2DS}}{\text{H}}}\\ =\sqrt{\frac{2\times 15,000\times 16}{31}}\\ =400\text{lb}\text{of}\text{beans}\end{array}$

EOQ calculated by multiplying 2, 15,000 and 16 and dividing the resultant with 31 and taking square root which gives 400 lb. of beans.

Hence, Economic Order Quantity is 400 lb. of beans.

Summary Introduction

To determine: The total annual holding cost.

Answer to Problem 49P

The total annual holding cost is $600.

Explanation of Solution

Given information:

$\begin{array}{c}\text{Annual}\text{demand,D}=75\times 200\\ =15,000\text{lb/year}\\ \text{Setup}\text{cost,S}=\$16\text{/order}\\ \text{Holding}\text{cost,H}=\$3\text{lb/year}\end{array}$

$\begin{array}{c}\text{Standard}\text{deviation,}\text{σ}=\text{15}\text{pounds/day}\\ \text{Lead}\text{time, L}=\text{4}\text{days}\\ \text{Number}\text{of}\text{working}\text{days}=\text{200}\end{array}$

Formula:

$\text{Annual}\text{holdingcost}=\frac{\text{Q}}{\text{2}}\times \text{H}$

Calculation annual holding cost:

$\begin{array}{c}\text{Annual}\text{holding}\text{cost}=\frac{\text{Q}}{\text{2}}\times \text{H}\\ \text{=}\frac{400}{2}\times \$3\\ =\$600\end{array}$

Annual holding cost is calculated by multiplying half of the economic order quantity which is 400 divided by 2 and multiplying with $3 which yields $600.

Hence, the annual holding cost is $600.

Summary Introduction

To determine: The annual setup cost.

Answer to Problem 49P

The annual setup cost is $530.33.

Explanation of Solution

Given information:

$\begin{array}{c}\text{Annual}\text{demand,D}=75\times 200\\ =15,000\text{lb/year}\\ \text{Setup}\text{cost,S}=\$16\text{/order}\\ \text{Holding}\text{cost,H}=\$3\text{lb/year}\end{array}$

$\begin{array}{c}\text{Standard}\text{deviation,}\text{σ}=\text{15}\text{pounds/day}\\ \text{Lead}\text{time, L}=\text{4}\text{days}\\ \text{Number}\text{of}\text{working}\text{days}=\text{200}\end{array}$

Formula:

$\text{Annual}\text{setup}\text{cost}=\frac{\text{D}}{\text{Q}}\text{×S}$

Calculation annual ordering cost:

$\begin{array}{c}\text{=}\frac{15,000}{400}\times 16\\ =\$600\end{array}$

Annual ordering cost is calculated by dividing 15,000 with 400 and multiplying the resultant with 16 which yields $600.

Hence, the annual ordering cost is $600.

Summary Introduction

To determine: The Reorder point (ROP).

Answer to Problem 49P

The Reorder point (ROP) is 369.99.

Explanation of Solution

Given information:

$\begin{array}{c}\text{Annual}\text{demand,D}=15,000\text{lb/year}\\ \text{Setup}\text{cost,S}=\$16\text{/order}\\ \text{Holding}\text{cost,H}=\$3\text{lb/year}\\ \text{Standard}\text{deviation,}\text{σ}=\text{15}\text{pounds/day}\end{array}$

$\begin{array}{c}\text{Lead}\text{time, L}=\text{4}\text{days}\\ \text{Number}\text{of}\text{working}\text{days}=\text{200}\\ \text{Stock}\text{out}\text{risk}=1\%\\ \text{Z}=2.33(99\%\text{service}\text{level})\end{array}$

Formula:

$\text{ROP}=\text{Average demand during the lead time (μ)+}\left(\text{Z}\times {\text{σ}}_{\text{dlt}}\right)$

Calculation of ROP:

$\begin{array}{c}\text{ROP}=\left(75\times 4\right)+2.33\times \left(\sqrt{4}\times 15\right)\\ =369.99\end{array}$

ROP is calculated by adding average demand during lead time (300) with the 69.9, which is the safety stock, which results in 369.99

Hence, ROP is 369.99.

Summary Introduction

To determine: The safety stock.

Answer to Problem 49P

Safety stock is 69.99.

Explanation of Solution

Given information:

$\begin{array}{c}\text{Annual}\text{demand,D}=15,000\text{lb/year}\\ \text{Setup}\text{cost,S}=\$16\text{/order}\\ \text{Holding}\text{cost,H}=\$3\text{lb/year}\\ \text{Standard}\text{deviation,}\text{σ}=\text{15}\text{pounds/day}\end{array}$

$\begin{array}{c}\text{Lead}\text{time, L}=\text{4}\text{days}\\ \text{Number}\text{of}\text{working}\text{days}=\text{200}\\ \text{Stock}\text{out}\text{risk}=1\%\\ \text{Z}=2.33(99\%\text{service}\text{level})\end{array}$

Formula:

$\text{Safety}\text{Stock,}\text{SS}={\text{Zσ}}_{\text{dlt}}$

Calculation of safety stock:

$\begin{array}{l}=2.33\times \left(\sqrt{4}\times 15\right)\\ =69.99\end{array}$ (1)

Safety stock is calculated by multiplying 2.33 and $\left(\sqrt{4}\times 15\right)$ which gives 69.99 as safety stock.

Hence, safety stock is 69.99.

Summary Introduction

To determine: The annual safety stock holding cost.

Answer to Problem 49P

The annual safety stock holding cost is $209.97.

Explanation of Solution

Given information:

$\begin{array}{c}\text{Annual}\text{demand,D}=15,000\text{lb/year}\\ \text{Setup}\text{cost,S}=\$16\text{/order}\\ \text{Holding}\text{cost,H}=\$3\text{lb/year}\\ \text{Standard}\text{deviation,}\text{σ}=\text{15}\text{pounds/day}\end{array}$

$\begin{array}{c}\text{Lead}\text{time, L}=\text{4}\text{days}\\ \text{Number}\text{of}\text{working}\text{days}=\text{200}\\ \text{Stock}\text{out}\text{risk}=1\%\\ \text{Z}=2.33(99\%\text{service}\text{level})\end{array}$

Formula:

$\text{Annual Safety}\text{Stock}\text{holding cost}={\text{Zσ}}_{\text{dlt}}\times \text{holding}\text{cost}$

Calculation of annual safety stock holding cost:

$\begin{array}{c}\text{Annual Safety}\text{Stock}\text{holding cost}=69.99\times \$3\\ =\$209.97\end{array}$

Annual safety stock holding cost holding cost is calculated by multiplying safety stock 69.99 with holding cost $3 which gives result as $209.97.

Hence, the annual safety stock holding cost is $209.97.

Summary Introduction

To determine: The safety stock.

Answer to Problem 49P

Safety stock is 61.61.

Explanation of Solution

Given information:

$\begin{array}{c}\text{Annual}\text{demand,D}=15,000\text{lb/year}\\ \text{Setup}\text{cost,S}=\$16\text{/order}\\ \text{Holding}\text{cost,H}=\$3\text{lb/year}\\ \text{Standard}\text{deviation,}\text{σ}=\text{15}\text{pounds/day}\end{array}$

$\begin{array}{c}\text{Lead}\text{time, L}=\text{4}\text{days}\\ \text{Number}\text{of}\text{working}\text{days}=\text{200}\\ \text{Stock}\text{out}\text{risk}=2\%\\ \text{Z}=2.054(98\%\text{service}\text{level})\end{array}$

Formula:

$\text{Safety}\text{Stock,}\text{SS}={\text{Zσ}}_{\text{dlt}}$

Calculation of safety stock:

$\begin{array}{l}=2.054\times \left(\sqrt{4}\times 15\right)\\ =61.62\end{array}$ (2)

Safety stock is calculated by multiplying 2.054 and $\left(\sqrt{4}\times 15\right)$ which gives 61.62 as safety stock.

On comparing equation (1) & (2), there is less safety stock is required when the stock out risk percent increases.

Hence, safety stock is 61.62 and it decreases.