Question

Find the optimum number of batches (to the nearest whole number) of an item that should be produced annually (in order to minimize cost) if 300,000 units are to be made, it costs $4 to store a unit for one year, and it costs $300 to set up the factory to produce each batch.

A. 32 batches

B 47 batches

C. 34 batches

D. 45 batches

Answer 1

Given that,

300,000 units are to be made.

It costs $4 to store a unit for one year, and it costs $300 to set up the factory to produce each batch.

Let,

The optimum number of batches be x.

Size of each batch is given by $=\frac{300 000}{x}$

Setup cost of each batch is 300x

Storage cost$=\frac{1}{2}×4×(\frac{300 000}{x})=\frac{600 000}{x}$

Total cost is given by

Total Cost$=C(X)=300x+\frac{600 000}{x}$

Optimum number of batches are given for C'(x)=0

$\frac{d}{dx}(300x+\frac{600,000}{x})=0\\300−\frac{600,000}{x^2}=0\\x^2=\frac{600,000}{300}\\x^2=2,000\\x=44.72$

(Only positive values are considered)

Rounding to the nearest integer, we get,

x=45

Therefore,

45 batches should be produced annually to optimize cost.

Therefore,

Correct answer is option D. 45 batches.