Question

A company produces and sells 211,600 boxes of t-shirts each year. Each production run has a fixed cost of $400 and an additional cost of $3 per box of t-shirts. To store a box for a full year costs $2. What is the optimal number of production runs the company should make each year? Do not include units with your answer.

Answer 1

Answer:$x=46$

Step-by-step explanation:

Given

Company Produces and sells 211600 boxes of T-shirt each year

Fixed cost =$ 400

Additional cost = $ 3 per box

storage cost =$ 2

let x be the no of production run

therefore

Holding cost per year $=holding\ cost\times average\ holding\ items=2\times \frac{211600}{2x}=\frac{211600}{x}$

$Yearly\ ordering\ cost=cost\ during\ each\ order\times number\ of\ order\ Placed\ per\ year$

yearly ordering cost$=400x+3\times \frac{211600}{x}$

Total cost C(x)$=\frac{211600}{x}+400x+3\times \frac{211600}{x}$

differentiate C(x) w.r.t to x we get

$\frac{\mathrm{d} C(x)}{\mathrm{d} x}=400-\frac{3\times 211600}{x^2}-\frac{1}{211600}$

Put $\frac{\mathrm{d} C(x)}{\mathrm{d} x}=0$ to get max/min value

$400-\frac{211600}{x^2}-\frac{3\times 211600}{x^2}=0$

$x^2=\frac{211600}{100}$

$x=\sqrt{2116}$

$x=46$

therefore 46 runs must be performed

Answer 2

The optimal number of production runs the company should make each year is 46

The given parameters are:

Fixed cost =$ 400

Additional cost = $ 3 per box

Storage cost =$ 2

Boxes of T-shirt = 211600

Represent the number of orders with x

So, the storage cost (S) per year is:

S = Storage cost * Average holding items

This gives

$S = 2 * \frac{211600}{2x}$

$S = \frac{211600}{x}$

The yearly ordering cost (Y) is:

Y = Fixed cost * Number of orders + Additional cost * Storage cost per year.

So, we have:

$Y = 400 * x + 3 * \frac{211600}x$

$Y = 400x + \frac{634800}x$

The total cost (T) is:

T = S + Y

So, we have:

$T = 400x + \frac{634800}x + \frac{211600}x$

$T = 400x + \frac{846400}x$

Differentiate

$T' = 400 - \frac{846400}{x^2}$

Set to 0

$400 - \frac{846400}{x^2} = 0$

Collect like terms

$\frac{846400}{x^2} = 400$

Divide through by 400

$\frac{2116}{x^2} = 1$

Take the square roots of both sides

$\frac{46}{x} = 1$

Cross multiply

$x = 46$

Hence, the optimal number of production runs the company should make each year is 46

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