   Chapter 10, Problem 53RE ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Inventory cost model A company needs to produce 288,000 items per year. Production costs are $1500 to prepare for a production run and$30 for each item produced. Inventory costs are $1.50 per year for each item stored. Find the number of items that should be produced in each run so that the total costs of production and storage are minimum. To determine To calculate: The number of items that should be produced in each run so that the total costs of production and storage are minimum if a company needs to produce 288,000 items per year. Production costs are$1500 for pre-production costs and $30 for each item produced and inventory costs are$1.50 per year for each item stored.

Explanation

Given Information:

A company needs to produce 288,000 items per year and production costs for the same are $1500 for pre-production and$30 for the production of each item and inventory costs are $1.50 per year for each item stored. Calculation: As it is provided that a company needs to produce 288,000 items per year and production costs for the same are$1500 for pre-production and $30 for the production of each item and inventory costs are$1.50 per year for each item stored.

Let x represent the number of items produced in each run.

The total production cost is:

P=Production cost of onr run+Production cost of each item

Thus, to produce 288,000 items, the production cost will be,

P=(288,000x)(1500)+(288,000)(30)

And,

The total storage cost is \$1.5 per year for each item stored.

Now, the total cost will be the number of items stored. Since, only half of the item will be stored as the other half will be in the production unit, so the total storage cost will be,

S=(x2)(1.5)

Thus, the total cost is:

C=(288,000x)(1500)+(288,000)(30)+(x2)(1.5)

Simplify the equations as follows:

C=(288,000x)(1500)+(288,000)(30)+(x2)(1.5)=432000000x+8640000+0.75x

Take out the first derivative of the equation by the power rule,

C=ddx(432000000x+8640000+0.75x)=ddx(432000000x)+ddx(8640000)+0.75ddx(x)=432000000x2+0.75

Put the value of C(x)=0,

0=432000000x2+0

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