Solve and Economic Order Quantity (EOQ) Problem
Case Study: Economic Order Quantity
The air filter manufacturer in problem (2) informs you that due to limited production capacity and high demand, he cannot ship your order at once, but instead will ship you 300 filters per day until the order is filled. Again, the company is open for business 20 days per month, 12 months per year and it takes 2 days for an order to arrive once it has been placed.
(a) What quantity do you order now?
Solution: This corresponds to the POQ model. We have the following data:

Input: 

Carrying cost, Cc = 
$2.00  
Setup or Ordering cost, Co = 
$10  
Annual Demand, D = 
36,000  
Annual days, months or weeks = 
240  
Demand rate, d = 
150.00  
Production rate, p = 
300 
The quantity to order is
\[{{Q}^{*}}=\sqrt{\frac{2DS}{H\left( 1\frac{d}{p} \right)}}=\sqrt{\frac{2\times 36,000\times 10}{2\left( 1\frac{150}{300} \right)}}=849\]
The following table summarizes the results:
Output: 

Qopt = 
849  
Total Cost, TC = 
$848.53  
Production run length = 
2.83  
Number of runs = 
42.43  
Maximum inventory = 
424.26  
Cycle Time = 
5.66 
(b) What is the maximum inventory level?
Solution: The maximum inventory level is
\[Q*\left( 1\frac{d}{p} \right)=849\left( 1\frac{150}{300} \right)=425\]
(c) Compute the total annual inventory costs.
Solution: The annual inventory costs are
\[IC=\frac{HQ*}{2}\left( 1\frac{d}{p} \right)=\frac{2\times 849}{2}\left( 1\frac{150}{300} \right)=\$424.26\]
(d) What is the reorder point?
Solution: The reorder point is
\[ROP = d\times L = 150\times 2 = 300\]
(Notice that this reorder point is valid for after the inventory has reached its maximum of 424.26)
4. Gentle Electric, which we discussed in class, has learned about a new company on the West Coast that has the following prices (I've included the 50cent unloading fee with the price.)
first 100 $485.50
next 100 $470.50
next 100 $450.50
Holding costs remain at 20% of the value of the inventory, maximum warehouse space is still 200 units, and each time you place an order it costs $25.
(a) Compute the average cost for if the order quantity is 285 units.
Solution: If \(Q = 285\), then we fall in the third price bracket, which means that the total cost is
\[TC=\frac{DS}{Q}+\frac{HQ}{2}+PD = 2\times \frac{D\times 25}{285}+450.50\times D\]
which means that the average cost is
\[AC=\frac{TC}{D}=\frac{50}{285}+450.50=450.6754\]
(b) Compute the annual ordering costs if the order quantity is 285 units.
Solution: In the case of the optimal order quantity, the ordering and holding costs are equal, which means that
\[OC=HC=\frac{HQ}{2}=\frac{0.20\times 450.50\times 285}{2}=\$12,839.25\]
(c) Compute the annual holding costs if the order quantity is 285 units.
Solution: For the EOQ, the holding and ordering costs are the same. In this case
\[OC=\$12,839.25\]
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