Question

Rocky Mountain Tire Center sells 12,000 ​go-cart tires per year. The ordering cost for each order is ​$35​, and the holding cost is 30​% of the purchase price of the tires per year. The purchase price is ​$20 per tire if fewer than 200 tires are​ ordered, ​$16 per tire if 200 or​ more, but fewer than 5,000​, tires are​ ordered, and ​$14 per tire if 5,000 or more tires are ordered.

a) How many tires should Rocky Mountain order each time it places an​ order? Rocky​ Mountain's optimal order quantity is ___units ​(enter your response as a whole​ number).

​b) What is the total cost of this​ policy?

Answer 1

Annual demand(D) = 12000 tires

Ordering cost (S) =$35

Holding cost(H) = 30% of purchase price

Order size Price per unit Holding cost(30% of price per unit)

0-200 20 6

200-5000 16 4.8

5000 or more 14 4.2

First find the minimum point for each price starting with the lowest price until feasible minimum point is located.This means until a minimum point falls in the quantity range for its price

Minimum point for price of $14 = Sqrt of (2DS/H)=Sqrt of [(2X12000X35)/4.2] = sqrt of 200000 = 447 tyres.Because an order size of 447 tyres will cost $16 rather than $14, 447 is not a minimum feasible point for $14 per unit.

Minimum point for price of $16 = Sqrt of (2DS/H) =Sqrt of [(2X12000X35)/4.8]= sqrt of 175000 = 418 tyres.This is feasible as it falls in the $16 per tyre range of 200-5000

Now the total cost for 418 tyres is computed and compred to the total cost of the minimum quantity needed to obtain price of $14 per tyre

Total cost for Q=418 is (Q/2)H + (D/Q)S + (PriceXD)

= [(418/2)4.8] + [(12000/418)35] + (16X12000)

= 1003.2 + 1004.78 + 192000

= $194007.98

The minimum quantity needed to obtain a price of $14 is 5000 units.So with order quantity(Q) = 5000 units,

Total cost = (Q/2)H + (D/Q)S + (PriceXD)

= [(5000/2)4.2] + [(12000/5000)35] + (14 x 12000)

= 10500 + 84 + 168000

= $178584

a) So Rocky mountain's optimal order quantity is 5000 units as it has the lowest total cost.so they should order 5000 tires each time

b) The total cost of ordering optimal order is $178584