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Rocky Mountain Tire Center sells 20,000 go cart tires per year. The ordering cost for each tire is \$40, and the holding cost is 20% of the purchase price of the tires per year. The purchase price is \$20 per tire if fewer than 500 tires are ordered, \$18 per tire if 500 or more-but fewer than 1,000 tires are ordered, and \$17 per tire if 1,000 or more tires are ordered. How many tires should Rocky Mountain order each time it places an order?

here's for everyone who had or needs this question answered and hasn't realized it has two answers!!!(
blame the jerk who wrote it!)

STEP 1:
For each discount, calculate a value for optimal order size Q*, using the following Equation.

Economic Order Quantity using equation (12.10) in text
Q* = √(2DS/IP)
Where,
Q= Quantity Ordered
D = Annual Demand = 20,000 units
S = Ordering or setup cost per order = \$40
P = Price per unit = \$20  (added this line)
H = Holding Cost per year  (I’ve shifted IP to the next line)
IP = percent (I) of unit price (P) where holding cost is 20% of purchase price => IP = 0.2

Q1* = √((2(20000)(40))/((.2)(20.00)))

Q1* = √400,000
Q1* = 632.45 ≈ 632 units per order

Q2 * = √((2(20000)(40))/((.2)(18.00)))
Q2 * = √444,444
Q2 * = 666.66666666 ≈ 667 units per order

Q3 * = √((2(20000)(40))/((.2)(17.00)))
Q3 * = √470,588
Q3 * = 685.994 ≈ 686 per order
Good.

STEP 2:
For any discount, if the order quantity is too low for the discount, adjust the order quantity upward to the lowest quantity that will qualify for the discount.
***Note that an order quantity as in step 1 that is greater than the range that would qualify it for a discount may be discarded

Calculated and Adjusted order quantities

Q1* 632 per order No adjustment required as 632 is outside the <500 range, but within the quantity break of 500-999 –
***Note that an order quantity as in step 1 that is greater than the range that would qualify it for a discount may be discarded.

Q2 *= 667 per order No adjustment required as 667 is within the quantity break of 500-999

Q3 *= 686 per order Adjusted to 1,000 per order

STEP 3:
Use the total cost equation TC=(D/Q) (S) + (Q/2) (H) + PD, compute a total cost for every Q* determined in STEPS 1 and 2. If we had to adjust Q3* upward because it was below the allowable quantity range, be sure to use the adjusted value for Q*.

Q1* = 632 -->
Q1* the holding cost (H) =IP or the percent (I) of unit price (P)
IP = (.2)(\$20) = \$4.00

TC =(D/Q) (S) + (Q/2) (H) + PD
TC =[(20,000)/632) (\$40) + (632/2) (\$4.00)] + (20,000)(\$20)
TC =(\$2,529.82) + (\$400,000)
TCfor Q1* = \$402,530

Q2 * = 667 -->
Q2* the holding cost (H) = IP or the percent (I) of unit price (P)
IP = (.2)(\$18) = \$3.60

TC =(D/Q) (S) + (Q/2) (H) + PD
TC =[(20,000/667) (\$40) + (667/2) (\$3.60)] + (20,000)(\$18)
TC =(\$2,400) + (\$360,000)
TCfor Q2* = \$362,400

Q3 * = 1,000 --adjusted -->
Q3* the holding cost (H) =IP or the percent (I) of unit price (P)
IP = (.2)(\$17) = \$3.40

TC =(D/Q) (S) + (Q/2) (H) + PD
TC =[(20,000/1,000)(\$40) + (1000/2)(\$3.4) ]+ (20,000)(\$17)
TC =(\$2500) + (\$340,000)
TC = \$342,500

Discard Q1* at 632 units because the unit price (purchase cost) is \$20 and the quantity of 632 is within the next quantity break range from 500 to 999 at a lower unit price (purchase cost) of \$18.

STEP 4:
Select the order quantity that has the lowest total cost, as computed in Step 3. It will be the quantity that will minimize the total inventory cost.

***When we simply look at the chart we get the impression that that Q3* ,where Q≥1000, would be the ideal amount, because there is a \$19,900 difference between itself and Q2* and the difference between Q3* and Q1* is \$60,029.

As the text states the major tradeoff when considering quantity discounts is between reduced product cost and increased holding cost.

The upper and lower limits of the range for Q* ≥ 1000 can range quite far apart and order quantities can range anywhere from 1000 units to 20,000 units per order.

Bearing in mind that holding cost does sometimes tend to increase when order quantity increases we made some extra calculations [***SEE EXCEL SPREADSHEET Problem 12.23]

As you can see from the chart Q≥1000 does have the lowest total annual cost when Q=1000 units, but as units increase so does holding cost. If we look at the Q* ≥ 1000 range we see that while the initial annual total cost is significantly lower than the others, as Q increases [significantly] so does the holding and setup cost until the point where the annual setup and holding cost for 20.000units is \$34,040 and the annual total cost is \$374,040. That is a \$ 31,540 difference from when Q= 1000 units.
Technically after 13,140 units [in the range Q*≥1000] holding costs rises to \$ \$22,338.00 and annual total cost is \$362,399; at value of \$362,400 the annual total cost of 13,140 units is the same as Q 2 *667 units [when range is 500-999 and discount price =\$18] as can be seen in the calculations below.
\$3.60 Holding Cost when amount of tires ordered is ≤ 500 tires but < 1000
(D ÷ Q)(S) = (Q ÷2)(H)
=(20,000/Q)(40.00) = (Q/2)(\$3.60)
=(2)(20,000)(40.00) = Q2 (\$3.60)
Q2 = [(2 ×20,000 ×40)/\$3.60] =
EOQ = √([(2 ×20,000 ×40)/\$3.60]) = √444,444 = 666.67
Q≈ 667 units
TC= \$362,400

If we look at our calculations we see that the range Q*≥1000 it is the ideal quantity amount when Q≥ 1000units ≤ 13140 units; afterward we lose a lot of cost savings.

Ultimately after all these extra simulated calculations (**SEE SPREADSHEET) we would argue that the range of Q* ≤ 500 tires but > 1000 TIRES ORDERED is the most concise range because the annual setup and holding cost per order will be between ≥\$2500 and \$2598 and the Annual Total Cost of the order will be between ≥\$362,500 and \$362,598. The range of Q* is ≥ 500 tires but < 1000 Tires Ordered may not have the lowest initial annual cost, but it does have the most constrained range of annual costs overall of each of the quantity level discounts, it does not incur high holding costs and satisfies the overall inventory model objective of minimizing total cost.

In addition, the optimal order quantity [when annual setup cost equals annual holding cost] of 667 units also lies within this range and it fits the criteria of the quantity discount. In summation, Rocky Mountain Tires could order ~667 tires every time it places an order.
Also, all the order quantities (Q*) that we calculated for Step 1 are in the 600's (632,667,689 respectively).

However, within the Q* ≥ 1000 range we see that the initial annual total cost is significantly lower than the others, which is why Rocky Mountain tires should also be open to placing their order quantities which are ≥ 1000.

Furthermore the quantity that Rocky Mountain tires will order depends on how much they need, how much of a yearly demand they have left, and how much money they want to save. Regardless of how much they order, we would not recommend they let their Q* go over 13,140 units or under 667 units as this would do nothing to minimize total cost.

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