Question

Rocky Mountain Tire Center sells 10,000 go-cart tires per year. The ordering cost for each order is​$40​,and the holding cost is 50​%of the purchase price of the tires per year. The purchase price is​ $21 per tire if fewer than 200 tires are​ ordered,​$19 per tire if 200 or​ more, but fewer than 8,000​ tires are​ ordered, and​ $13 per tire if 8,000 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?

Total annual cost of ordering optimal order size = $_____​(round your response to the nearest whole​ number).

Answer 1

Annual demand(D) = 10000 tires

Ordering cost (S) =$40

Holding cost(H) = 50% of purchase price

Order size Price per unit Holding cost(50% of price per unit)

0-200 21 10.5

200-8000 19 9.5

8000 or more 13 6.5

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 $13 = Sqrt of (2DS/H)=Sqrt of [(2X10000X40)/6.5] = √(800000/6.5) = sqrt of 123076.923 = 351 tyres.Because an order size of 351 tyres will cost $19 rather than $13, 351 is not a minimum feasible point for $13 per unit.

Minimum point for price of $19 = Sqrt of (2DS/H) =Sqrt of [(2X10000X40)/9.5]= √(800000/9.5) = sqrt of 44210.52631 = 290 tyres.This is feasible as it falls in the $19 per tyre range of 200-8000

Now the total cost for 290 tyres is computed and compred to the total cost of the minimum quantity needed to obtain price of $13 per tyre

Total cost for Q=290 is (Q/2)H + (D/Q)S + (PriceXD)

= [(290/2)9.5] + [(10000/290)40] + (19X10000)

= 1377.5 + 1379.31 + 190000

= $192756.81

The minimum quantity needed to obtain a price of $13 is 8000 units.So with order quantity(Q) = 8000 units,

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

= [(8000/2)6.5] + [(10000/8000)40] + (13 x 10000)

= 26000 + 50 + 130000

= $156050

a) So Rocky mountain's optimal order quantity is 8000 units as it has the lowest total cost.so they should order 8000 tires each time

b) The total cost of ordering optimal order is $156050