Chapter 12, Problem 30P

a)

Summary Introduction

To determine: The daily demand of the product.

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 30P

The daily demand of the product is 32.

Explanation of Solution

Given information:

$\begin{array}{c}\text{Annual}\text{demand,D}=\text{8,000}\\ \text{Daily}\text{production}\text{rate,p}=\text{200}\\ \text{Setup}\text{cost,S}=\text{120}\\ \text{Holding}\text{cost,H}=\text{50}\end{array}$

$\begin{array}{c}\text{Production}\text{quantity,Q}=\text{400}\\ \text{Number}\text{of}\text{working}\text{days}=\text{250}\end{array}$

Calculation of daily demand of the product:

$\begin{array}{c}\text{Daily}\text{demand,d}=\frac{\text{D}}{250}\\ =\frac{8,000}{250}\\ =32\end{array}$

The daily demand is calculated by dividing the annual demand, 8000 with the number of working days, 250 which gives the result as 32 units.

Hence, the daily demand is 32 units.

b)

Summary Introduction

To determine: The number of day production must be carried out.

Answer to Problem 30P

The number of day production must be carried out is 2days.

Explanation of Solution

Given information:

$\begin{array}{c}\text{Annual}\text{demand,D}=\text{8,000}\\ \text{Daily}\text{production}\text{rate,p}=\text{200}\\ \text{Setup}\text{cost,S}=\text{120}\\ \text{Holding}\text{cost,H}=\text{50}\end{array}$

$\begin{array}{c}\text{Production}\text{quantity,Q}=\text{400}\\ \text{Number}\text{of}\text{working}\text{days}=\text{250}\end{array}$

Calculation of number of days in production runs:

$\begin{array}{c}\text{Number}\text{of}\text{production}\text{days}=\frac{\text{Q}}{\text{p}}\\ =\frac{400}{200}\\ =2\end{array}$

Number of days production must be carried out is calculated by dividing the production quantity, (400) with production rate (200) which gives 2 days.

Hence, the number of day production must be carried out is 2days.

c)

Summary Introduction

To determine: The number production runs per year and annual setup cost.

Answer to Problem 30P

The number of day production runs per year is 20 and annual setup cost is $2,400.

Explanation of Solution

Given information:

$\begin{array}{c}\text{Annual}\text{demand,D}=\text{8,000}\\ \text{Daily}\text{production}\text{rate,p}=\text{200}\\ \text{Setup}\text{cost,S}=\text{120}\\ \text{Holding}\text{cost,H}=\text{50}\end{array}$

$\begin{array}{c}\text{Production}\text{quantity,Q}=\text{400}\\ \text{Number}\text{of}\text{working}\text{days}=\text{250}\end{array}$

Calculation of number of production runs per year and annual setup cost:

Number of production runs per year:

$\begin{array}{c}\text{Number}\text{of}\text{production}\text{runs}\text{per}\text{year}=\frac{\text{D}}{\text{Q}}\\ =\frac{8,000}{400}\\ =20\end{array}$

Number of production runs per year is calculated by dividing the demand (8000) with production quantity (400) which gives 20.

Annual setup cost:

$\begin{array}{c}\text{Annual setup cost}=20\times \$120\\ =\$2,400\end{array}$

Annual setup cost is calculated by multiplying number of production runs per year (20) with setup cost (120) which gives $2,400 as annual setup cost.

Hence, the number of production runs per year is 20 and annual setup cost is $2,400.

d)

Summary Introduction

To determine: The maximum and average inventory levels.

Answer to Problem 30P

The maximum and average inventory levels are 336 and 168 respectively.

Explanation of Solution

Given information:

$\begin{array}{c}\text{Annual}\text{demand,D}=\text{8,000}\\ \text{Daily}\text{production}\text{rate,p}=\text{200}\\ \text{Setup}\text{cost,S}=\text{120}\\ \text{Holding}\text{cost,H}=\text{50}\end{array}$

$\begin{array}{c}\text{Production}\text{quantity,Q}=\text{400}\\ \text{Number}\text{of}\text{working}\text{days}=\text{250}\end{array}$

Calculation of maximum inventory and average inventory level:

Maximum inventory level:

$\begin{array}{c}\text{Maximum}\text{inventory}=\text{Q}\left(\text{1}-\frac{\text{d}}{\text{p}}\right)\\ =400\left(1-\frac{32}{200}\right)\\ =336\end{array}$

Maximum inventory is calculated by multiplying production quantity (400) with $\left(1-\frac{32}{200}\right)$ which yields 336.

Average inventory level:

$\begin{array}{c}\text{Average}\text{inventory}\text{level}=\frac{\text{Maximum}\text{inventory}}{\text{2}}\\ =\frac{336}{2}\\ =168\end{array}$

Average inventory is calculated by dividing maximum inventory level, 336 by 2 which gives168.

Hence, the maximum and average inventory levels are 336 and 168 respectively.

e)

Summary Introduction

To determine: The total annual setup cost and holding cost.

Answer to Problem 30P

The total annual setup cost and holding cost is $8,400 and $2400 respectively.

Explanation of Solution

Given information:

$\begin{array}{c}\text{Annual}\text{demand,D}=\text{8,000}\\ \text{Daily}\text{production}\text{rate,p}=\text{200}\\ \text{Setup}\text{cost,S}=\text{120}\\ \text{Holding}\text{cost,H}=\text{50}\end{array}$

$\begin{array}{c}\text{Production}\text{quantity,Q}=\text{400}\\ \text{Number}\text{of}\text{working}\text{days}=\text{250}\end{array}$

Calculation of total annual setup and holding cost:

$\begin{array}{c}\text{Total}\text{holding}\text{cost}=\text{Average}\text{inventory}\times \text{holding}\text{cost}\\ =\text{168}\times \text{50}\\ =\text{\$8400}\end{array}$

Total holding cost is calculated by multiplying average inventory (168) with holding cost (50) which yields $8,400.

$\begin{array}{c}\text{Total}\text{setup}\text{cost}=\text{Number}\text{of}\text{production}\text{runs}\text{per}\text{year}\times \text{Setup}\text{cost}\\ =\text{20}\times \text{120}\\ =\text{\$2,400}\end{array}$

$\begin{array}{c}\text{Total}\text{cost}=\text{Total}\text{holding}\text{cost}+\text{Total}\text{setup}\text{cost}\\ =\$8400+\$2400\\ =\$10,800\end{array}$ (1)

Total setup cost is calculated by multiplying number of production runs (20) with setup cost (120) which yields $2,400.

Hence, the total annual setup cost and holding cost is $8,400 and $2400 respectively.

f)

Summary Introduction

To determine: The number of refrigerators to be produced in each production line to minimize the total annual inventory cost and the net saving in inventory.

Answer to Problem 30P

The number of refrigerators to be produced in each production line to minimize the total annual inventory cost is 37.1 and the net saving in inventory is $1,820.

Explanation of Solution

Given information:

$\begin{array}{c}\text{Annual}\text{demand,D}=\text{8,000}\\ \text{Daily}\text{production}\text{rate,p}=\text{200}\\ \text{Setup}\text{cost,S}=\text{120}\\ \text{Holding}\text{cost,H}=\text{50}\end{array}$

$\begin{array}{c}\text{Production}\text{quantity,Q}=\text{400}\\ \text{Number}\text{of}\text{working}\text{days}=\text{250}\end{array}$

Calculation of number of refrigerators produced in each production run:

$\begin{array}{c}\text{Q}=\sqrt{\frac{\text{2DS}}{\text{H}\left(\text{1}-\frac{\text{d}}{\text{p}}\right)}}\\ =\sqrt{\frac{\text{2}\times 8\text{,000}\times 12\text{0}}{50\left(\text{1}-\frac{32}{200}\right)}}\\ =213.81\text{refrigerators}\end{array}$

Number of refrigerators produced in each production run is calculated by dividing $\text{2}\times 8\text{,000}\times 12\text{0}$ with $50\left(\text{1}-\frac{32}{200}\right)$ and taking square root which yields 213.81 refrigerators.

Calculation of holding and setup cost:

Maximum inventory level:

$\begin{array}{c}\text{Maximum}\text{inventory}=\text{Q}\left(\text{1}-\frac{\text{d}}{\text{p}}\right)\\ =213.81\left(1-\frac{32}{200}\right)\\ =179.6004\end{array}$

Maximum inventory is calculated by multiplying production quantity (213.81) with $\left(1-\frac{32}{200}\right)$ which yields 179.6004.

Average inventory level:

$\begin{array}{c}\text{Average}\text{inventory}\text{level}=\frac{\text{Maximum}\text{inventory}}{\text{2}}\\ =\frac{179.6004}{2}\\ =89.8002\end{array}$

Holding cost:

$\begin{array}{c}\text{Total}\text{holding}\text{cost}=\text{Average}\text{inventory}\times \text{holding}\text{cost}\\ =\text{89}\text{.8002}\times \text{50}\\ =\text{\$4490}\end{array}$

Total holding cost is calculated by multiplying average inventory (89.6002) with holding cost (50) which yields $4,490.

Number of production runs per year:

$\begin{array}{c}\text{Number}\text{of}\text{production}\text{runs}\text{per}\text{year}=\frac{\text{D}}{\text{Q}}\\ =\frac{8,000}{213.81}\\ =37.41\end{array}$

Number of production runs per year is calculated by dividing the demand (8000) with production quantity (213.81) which gives 37.41.

Setup cost:

$\begin{array}{c}\text{Total}\text{setup}\text{cost}=\text{Number}\text{of}\text{production}\text{runs}\text{per}\text{year}\times \text{Setup}\text{cost}\\ =\text{37}\text{.42}\times \text{120}\\ =\text{\$4,490}\end{array}$

Total cost:

$\begin{array}{c}\text{Total}\text{cost}=\text{Total}\text{holding}\text{cost}+\text{Total}\text{setup}\text{cost}\\ =\$4490+\$44900\\ =\$8,980\end{array}$ (2)

Total cost is the sum of holding cost and set up cost which is sum of $4,490 and $4,490 which gives $8,980.

Total saving:

$\begin{array}{c}\text{Savings}=\$10,800-\$8,980\\ =\$1,820\end{array}$

Total savings in cost is calculated by taking the difference of $10,800 and $8,980 which gives $1,820.

Hence, the number of refrigerators to be produced in each production line to minimize the total annual inventory cost is 37.1 and the net saving in inventory is $1,820.