Minimizing Inventory Costs.

Question

A clothing store sells 2400 pairs of white socks per year. It costs $0.04 to store one pair of white socks for one year. To reorder, there is a fixed cost of $0.75, plus $0.02 for each pair of socks. How many times per year should the store order white socks and in what lot size, in order to minimize inventory costs?

Answer 1

Given the information in the problem statement, we must assume that the rate of consumption is constant at 2400 pairs per year, throughout the year, or 2400 units per unit time which is a year. If there are S pairs of socks in store to begin with, then at time t (in years, or a fraction thereof), there are

( S – 2400*t ) 

units. As the cost of storage is proportional to this, as well as time ("$0.04 to store one pair of white socks for one year"), in a small incremental time, dt, it is (in dollars)

( S – 2400*t )*0.04*dt

In a best-case scenario, the stocking shelves are replenished just as they become empty, n times a year, with n TBD. Therefore, once replenished, there would immediately be one n-th the amount needed per year on the storage shelves, such that

S = 2400 / n

The cost, in dollars, of storing per restocking period (1/n year) is then found by integration:

int_(t=0)^(t=1/n)(( 2400/n – 2400*t )*0.04)*dt = 0.04*2400*(1/n² - 0.5/n²) = 48/n²

The cost of reordering per restocking period is, in dollars,

(2400/n)*0.02+0.75 = 48/n+0.75

Therefore, the total cost per year is

n*(48/n²+48/n+0.75) = 48/n + 48 + 0.75n

We minimize this by differentiating w.r.t. n and equating the result to zero, then solving for n:

-48/n² + 0.75 = 0, n = sqrt(48/0.75) = 8,

exactly.

Answer: To minimize cost, the store should order (2400/8=) 300 socks 8 times per year.

Answer 2

I will help you get started on this problem. 

 

As all pairs of white socks must be in stock at the clothing store before they're sold, the total storage cost (SC) for a one year supply of white socks  =  2,400 * $0.04  =  $96 

 

Also, the total fixed cost (FC) for this one year supply of white socks, with regard to EACH of the supply re-orders, becomes = $0.75 + $0.02(n) , (where n = # of pairs of white socks) . 

 

You can use this information to compare the inventory costs of a 2,400 white sock pair-lot size order to the inventory costs in other scenarios (i.e. 3 - 800 lot size orders, 4 - 600 lot size orders, etc.)  

 

GOOD LUCK !