Question

Rocky Mountain Tire Center sells 14,000 ​go-cart tires per year. The ordering cost for each order is $35, and the holding cost is 40​% 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 $18 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 _____ ?

b) What is the total cost of inventory?

Answer 1

Annual demand(D) = 14000 tires

Ordering cost (S) =$35

Holding cost(H) = 40% of purchase price

Order size Price per unit Holding cost(40% of price per unit)

0-200 21 8.4

200-8000 19 7.6

8000 or more 18 7.2

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 $18 = √(2DS/H)= √[(2X14000X35)/7.2] = √(980000/7.2) = √136111.1111 = 369 tyres.Because an order size of 369 tyres will cost $19 rather than $18, 369 is not a minimum feasible point for $18 per unit.

Minimum point for price of $19 = √(2DS/H) =√[(2X14000X35)/7.6]= √(980000/7.6) = √128947.3684 = 359 tyres.This is feasible as it falls in the $19 per tyre range of 200-8000

Now the total cost for 359 tyres is computed and compred to the total cost of the minimum quantity needed to obtain price of $18 per tyre

Total cost for Q=359 is (Q/2)H + (D/Q)S + (PriceXD)

= [(359/2)7.6] + [(14000/359)35] + (19X14000)

= 1364.2 + 1364.90 + 266000

= $268729.10

The minimum quantity needed to obtain a price of $18 is 8000 units.So with order quantity(Q) = 8000 units,

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

= [(8000/2)7.2] + [(14000/8000)35] + (18 x 14000)

= 28800 + 61.25 + 252000

= $280861.25

a) So Rocky mountain's optimal order quantity is 359 units as it has the lowest total cost.so they should order 359 tires each time

b) The total cost of ordering optimal order is $268729.10