## Tyre Company

Posted by alohablue on June 28, 2010 – 11:24 am

A tyre company in Namibia buys 2500 tyres per year from suppliers at an average N$500 per tyre.

The annual storage cost is about 15% of the purchase price. The ordering cost is N$250 per order.

How many tyres should they order each time???

- Alberto Engelbrecht

June 28th, 2010 at 3:46 pm

Assuming that the tyre sales, storage charges etc. can be treated in a continuous manner, and that the stock is replaced just in time.

Let the number of tyres/year turned over = R = 2500

Consider a batch of size B. The number of batches ordered per year N = R/B. Assuming the stock reduces to 0 and then is immediately reloaded with the new batch, then the average stock level will be B/2.

The cost of storing one tyre for a year 0.15*500 = 75.

The cost of ordering a batch is 250+500B.

The total cost of buying and storing tyres over a year

C = N(250+500B) + 75B/2. But B = R/N, so

C = 250N +500R + 37.5R/N

So dC/dN = 250 – 37.5R/N² = 0 at optimum =>

N² = 37.5 R/250 = 375, so N = 19.36…

The corresponding batch size is 129.099…

N = 19 => B = 131.5789…

N = 20 => B = 125 exactly.

The batch size and the number of orders must both be integers (assuming that you can’t split over year ends). So 20 batches of 125 should do the trick.

That the number aren’t nice worries me.

June 28th, 2010 at 3:53 pm

If you can split orders of year ends, then B = 129 and N = 2500/129 = 19.3798…

June 28th, 2010 at 4:43 pm

And before anyone starts: “yes” I’ve taken a kazillion liberties with reality.

June 29th, 2010 at 5:30 am

Pedantic mode on. For some reason I thought I was supposed to be finding the number of batches per year, rather than the size of each batch. I should have done:

The total cost of buying and storing tyres over a year

C = N(250+500B) + 75B/2. But N = R/B, so

C = 250R/B + 500R + 37.5B

So dC/dB = -250R/B² + 37.5 = 0 at optimum =>

B² = 250R/37.5 => B = 129.099..

Therefore chooose B = 129 (and tough that that causes different numbers of orders per year).

July 2nd, 2010 at 9:21 am

I just consider the cost of storage and the cost of the order together for 2500 tyres is 500/tyre.

I just got the answer: 9 times order with original cost of the tyre is 434.

(2500*434*(1 + 0.15) + 2250)/2500 = 500.

Here the order per batch is not the same quantity.

July 2nd, 2010 at 4:17 pm

Hi Venkat. I think that you have misunderstood the question. The tyres cost 500 to buy plus an overhead of 250 per order plus the storage overhead.

With a batch size of 129, the average overall cost of a single tyre is 500 + 250/129 + 37.5*129/2500 ≈ 503.87