Chapter 13, Problem 10P

A chemical firm produces sodium bisulfate in 100-pound bags. Demand for this product is 20 tons per day. The capacity for producing the product is 50 tons per day. Setup costs $100, and storage and handling costs are $5 per ton a year. The firm operates 200 days a year.

(Note: 1 ton = 2,000 pounds.)

a. How many bags per nm are optimal?

b. What would the average inventory be for this lot size?

c. Determine the approximate length of a production run, in days.

d. About how many runs per year would there be?

e. How much could the company save annually if the setup cost could be reduced to $25 per run?

Summary Introduction

To determine: The number of bags per runs that would be optimal.

Introduction: Inventory is a stock or store of goods. Every company store lots of goods as inventory which will be used during replenishment periods. Management of inventory is so much essential to manage cost and also to reduce cost.

Answer to Problem 10P

The number of bags per runs that would be optimal is 10,328 bags.

Explanation of Solution

Given information:

$\begin{array}{c}\text{D}=400\text{bags}\text{per}\text{year}\times 200\text{days}\text{per}\text{year}\\ =80,000\text{bags/year}\\ \text{Holding cost, H}=\frac{\text{\$5/ton/year}}{20\text{bags}\text{per}\text{ton}}\\ =0.25\text{/bag/year}\end{array}$

$\begin{array}{c}\text{Demand}\text{for}\text{product,u}=\frac{\text{20}\text{tons}\text{per}\text{day×2000}\text{pounds}\text{per}\text{ton}}{\text{100}\text{pounds}\text{per}\text{bag}}\\ =400\text{bags}\text{per}\text{year}\\ \text{capacity,p}=\frac{\text{50}\text{tons}\text{per}\text{day}\times \text{2000}\text{pounds}\text{per}\text{ton}}{\text{100}\text{pounds}\text{per}\text{bag}}\\ =1,000\text{bags}\text{per}\text{year}\end{array}$

$\text{Setup cost,S}=\$100$

Formula:

${\text{Q}}_{\text{p}}=\sqrt{\frac{\text{2DS}}{\text{H}}}\times \sqrt{\frac{\text{p}}{\text{p}-\text{u}}}$

Calculation of number of bags per runs that would be optimal:

$\begin{array}{c}{\text{Q}}_{\text{o}}=\sqrt{\frac{\text{2}\times 80\text{,000}\times 100}{0.25}}\times \sqrt{\frac{1,000}{1,000-400}}\\ =10,328\text{bags}\end{array}$

The number of bags per runs that would be optimal is calculated by dividing the product of 2 and 80,000 and 100 with 0.25 and taking square root which gives the resultant value as 8,000 bags and 1,000 is divided with the difference of 1,000 and 400 and square root is taken which gives 1.29, Both values are multiplies which gives the optimal run size as 10,328 bags.

Hence, the number of bags per runs that would be optimal is 10,328 bags.

Summary Introduction

To determine: The average inventory.

Introduction: Inventory is a stock or store of goods. Every company store lots of goods as inventory which will be used during replenishment periods. Management of inventory is so much essential to manage cost and also to reduce cost.

Answer to Problem 10P

The average inventory is 3098.4 bags.

Explanation of Solution

Given information:

$\begin{array}{c}\text{D}=400\text{bags}\text{per}\text{year}\times 200\text{days}\text{per}\text{year}\\ =80,000\text{bags/year}\\ \text{Holding cost, H}=\frac{\text{\$5/ton/year}}{20\text{bags}\text{per}\text{ton}}\\ =0.25\text{/bag/year}\end{array}$

$\begin{array}{c}\text{Demand}\text{for}\text{product,u}=\frac{\text{20}\text{tons}\text{per}\text{day×2000}\text{pounds}\text{per}\text{ton}}{\text{100}\text{pounds}\text{per}\text{bag}}\\ =400\text{bags}\text{per}\text{year}\\ \text{capacity,p}=\frac{\text{50}\text{tons}\text{per}\text{day}\times \text{2000}\text{pounds}\text{per}\text{ton}}{\text{100}\text{pounds}\text{per}\text{bag}}\\ =1,000\text{bags}\text{per}\text{year}\end{array}$

$\text{Setup cost,S}=\$100$

Formula:

$\text{Average}\text{inventory}=\frac{{\text{I}}_{\text{max}}}{2}$

Calculation of average inventory:

$\begin{array}{c}{\text{I}}_{\text{max}}=\frac{{\text{Q}}_{\text{p}}}{\text{p}}\left(\text{p}-\text{u}\right)\\ =\frac{10,328}{1,000}\left(1,000-400\right)\\ =6,196.8\end{array}$

$\begin{array}{c}\text{Average}\text{inventory}=\frac{6,196.8}{2}\\ =3,098.4\text{bags}\end{array}$

The average inventory is calculated by dividing 6,196 by 2 which yields 3,098.4 bags.

Hence, the average inventory is 3098.4 bags.

Summary Introduction

To determine: The approximate length of production run in days.

Introduction: Inventory is a stock or store of goods. Every company store lots of goods as inventory which will be used during replenishment periods. Management of inventory is so much essential to manage cost and also to reduce cost.

Answer to Problem 10P

The approximate length of production run in days is 10.33 days.

Explanation of Solution

Given information:

$\begin{array}{c}\text{D}=400\text{bags}\text{per}\text{year}\times 200\text{days}\text{per}\text{year}\\ =80,000\text{bags/year}\\ \text{Holding cost, H}=\frac{\text{\$5/ton/year}}{20\text{bags}\text{per}\text{ton}}\\ =0.25\text{/bag/year}\end{array}$

$\begin{array}{c}\text{Demand}\text{for}\text{product,u}=\frac{\text{20}\text{tons}\text{per}\text{day×2000}\text{pounds}\text{per}\text{ton}}{\text{100}\text{pounds}\text{per}\text{bag}}\\ =400\text{bags}\text{per}\text{year}\\ \text{capacity,p}=\frac{\text{50}\text{tons}\text{per}\text{day}\times \text{2000}\text{pounds}\text{per}\text{ton}}{\text{100}\text{pounds}\text{per}\text{bag}}\\ =1,000\text{bags}\text{per}\text{year}\end{array}$

$\text{Setup cost,S}=\$100$

Calculation of approximate length of production runs in days:

$\begin{array}{c}\text{Run}\text{length}=\frac{{\text{Q}}_{\text{p}}}{\text{p}}\\ =\frac{10,328}{\text{1,000}}\\ =\text{10}\text{.33days}\end{array}$

The approximate length of production runs in days is calculated by dividing 10,328 by 1,000 which yields 10.33 days.

Hence, approximate length of production runs in days is 10.33 days.

Summary Introduction

To determine: The number of runs per year.

Introduction: Inventory is a stock or store of goods. Every company store lots of goods as inventory which will be used during replenishment periods. Management of inventory is so much essential to manage cost and also to reduce cost.

Answer to Problem 10P

The number of runs per year is 7.55 runs per year.

Explanation of Solution

Given information:

$\begin{array}{c}\text{D}=400\text{bags}\text{per}\text{year}\times 200\text{days}\text{per}\text{year}\\ =80,000\text{bags/year}\\ \text{Holding cost, H}=\frac{\text{\$5/ton/year}}{20\text{bags}\text{per}\text{ton}}\\ =0.25\text{/bag/year}\end{array}$

$\begin{array}{c}\text{Demand}\text{for}\text{product,u}=\frac{\text{20}\text{tons}\text{per}\text{day×2000}\text{pounds}\text{per}\text{ton}}{\text{100}\text{pounds}\text{per}\text{bag}}\\ =400\text{bags}\text{per}\text{year}\\ \text{capacity,p}=\frac{\text{50}\text{tons}\text{per}\text{day}\times \text{2000}\text{pounds}\text{per}\text{ton}}{\text{100}\text{pounds}\text{per}\text{bag}}\\ =1,000\text{bags}\text{per}\text{year}\end{array}$

$\text{Setup cost,S}=\$100$

Calculation of number of runs per year:

$\begin{array}{c}\text{Number}\text{of}\text{runs}=\frac{\text{D}}{\text{Q}}\\ =\frac{80,000}{10,328}\\ =7.75\text{runs/year}\end{array}$

The number of runs per year is calculated by dividing demand of 80,000 with 10,328 which gives 7.75 runs per year.

Hence, the number of runs per year is 7.75 runs per year.

Summary Introduction

To determine: The annual saving by the company.

Introduction: Inventory is a stock or store of goods. Every company store lots of goods as inventory which will be used during replenishment periods. Management of inventory is so much essential to manage cost and also to reduce cost.

Answer to Problem 10P

The annual savings by the company is $774.59 per year.

Explanation of Solution

Given information:

$\begin{array}{c}\text{D}=400\text{bags}\text{per}\text{year}\times 200\text{days}\text{per}\text{year}\\ =80,000\text{bags/year}\\ \text{Holding cost, H}=\frac{\text{\$5/ton/year}}{20\text{bags}\text{per}\text{ton}}\\ =0.25\text{/bag/year}\end{array}$

$\begin{array}{c}\text{Demand}\text{for}\text{product,u}=\frac{\text{20}\text{tons}\text{per}\text{day×2000}\text{pounds}\text{per}\text{ton}}{\text{100}\text{pounds}\text{per}\text{bag}}\\ =400\text{bags}\text{per}\text{year}\\ \text{capacity,p}=\frac{\text{50}\text{tons}\text{per}\text{day}\times \text{2000}\text{pounds}\text{per}\text{ton}}{\text{100}\text{pounds}\text{per}\text{bag}}\\ =1,000\text{bags}\text{per}\text{year}\end{array}$

$\text{Setup cost,S}=\$25$

Calculation of annual saving by the company:

$\begin{array}{c}{\text{Q}}_{\text{o}}=\sqrt{\frac{\text{2}\times 80\text{,000}\times 25}{0.25}}\times \sqrt{\frac{1,000}{1,000-400}}\\ =5,164\text{bags}\end{array}$

The number of bags per runs that would be optimal is calculated by dividing the product of 2 and 80,000 and 25 with 0.25 and taking square root which gives the resultant value as 4,000 bags and 1,000 is divided with the difference of 1,000 and 400 and square root is taken which gives 1.29, Both values are multiplies which gives the optimal run size as 5,164 bags.

$\begin{array}{c}{\text{I}}_{\text{max}}=\frac{{\text{Q}}_{\text{p}}}{\text{p}}\left(\text{p}-\text{u}\right)\\ =\frac{5,164}{1,000}\left(1,000-400\right)\\ =3098.4\text{bags}\end{array}$

Maximum inventory is calculated by dividing 5,164 with 1000 and multiplying the resultant with the difference of 1000 and 400 which yields 3098.4 bags.

$\begin{array}{c}\text{TC}=\left(\frac{{\text{I}}_{\text{max}}}{2}\right)\text{H}+\left(\frac{\text{D}}{\text{Q}}\right)\text{S}\\ =\left(\frac{3,098.4}{2}\right)0.25+\left(\frac{80,000}{5,164}\right)25\\ =387.30+\mathrm{387.3.}\\ =\$774.60\end{array}$

The cost is calculated by computing the ordering and carrying cost. The ordering cost is calculated by multiplying half of the I_max with holding cost, 0.25 which results in $387.30. The carrying cost is calculated by dividing demand, 80,000 with EOQ, 5,164 and multiplying the resultant with 25 which gives $387.30. The sum of both cost accounts to $774.60.

Total cost when the setup cost is $100.

$\begin{array}{c}\text{TC}=\left(\frac{{\text{I}}_{\text{max}}}{\text{2}}\right)\text{H}+\left(\frac{\text{D}}{\text{Q}}\right)\text{S}\\ =\left(\frac{6,196.8}{2}\right)0.25+\left(\frac{80,000}{10,328}\right)100\\ =\$774.60+\$774.59\\ =\$1,549.19\end{array}$

The cost is calculated by computing the ordering and carrying cost. The ordering cost is calculated by multiplying half of the I_max with holding cost, 0.25 which results in $774.60. The carrying cost is calculated by dividing demand, 80,000 with EOQ, 10,328 and multiplying the resultant with 100 which gives $774.59. The sum of both cost accounts to $1,549.19.

Annual savings

$\begin{array}{c}\text{Savings}=\text{\$1,549}\text{.19}-\text{\$774}\text{.60}\\ =\text{\$774}\text{.59/year}\end{array}$

The savings in cost is the difference between $1,549 and $774.60 which gives $774.59.

Hence, the annual savings by the company is $774.59 per year.