**1. **Production Order Quantity Model

**2. **Production Order Quantity Model In EOQ Model, We assumed that the entire order was received at one time.
However, Some Business Firms may receive their orders over a period of time.

**3. **Production Order Quantity Model Such cases require a different inventory model.
Here, we take into account the daily production rate and daily demand rate.

**4. **Production Order Quantity Model

**5. **Production Order Quantity Model Since this model is especially suitable for production environments, It is called Production Order Quantity Model.
Here, we use the same approach as we used in EOQ model.
Lets define the following:

**6. **Production Order Quantity Model p: Daily Production rate (units / day)
d: Daily demand rate (units / day)
t: Length of the production in days.
H: Annual holding cost per unit

**7. **Production Order Quantity Model Average Holding Cost = (Average Inventory) . H
= (Max. Inventory / 2) . H

**8. **Production Order Quantity Model In the period of production (until the end of each t period):
Max. Inventory = (Total Produced) ? (Total Used)
= p.t - d.t

**9. **Production Order Quantity Model Here, Q is the total units that are produced.
Therefore,
Q = p.t t = Q/p

**10. **Production Order Quantity Model If we replace the values of t in the Max. Inventory formula:
Max. Inventory = p (Q/p) - d (Q/p) = Q - dQ/p = Q (1 ? d/p)

**11. **Production Order Quantity Model Annual Holding Cost = (Max. Inventory / 2) . H = Q/2 (1 ? d/p) . H
Annual Setup Cost = (D/Q) . S

**12. **Production Order Quantity Model Now we will set
Annual Holding Cost = Annual Setup Cost
Q/2 (1 ? d/p) . H = (D/Q) . S

**13. **Production Order Quantity Model

**14. **Production Order Quantity Model This formula gives us the optimum production quantity for the Production Order Quantity Model.
It is used when inventory is consumed as it is produced.

**15. **Backorder Inventory Model In this model, we assume that stock outs (and backordering) are allowed.

**16. **Backorder Inventory Model In addition to previous assumptions, we assume that sales will not be lost due to a stock out.
Because, we will back order any demand that can not be fulfilled.

**17. **Backorder Inventory Model B: Backordering cost per unit per year
b: The amount backordered at the time the next order arrives
Q ? b: Remaining units after the backorder is satisfied

**18. **Backorder Inventory Model

**19. **Backorder Inventory Model Total Annual Cost = Annual Setup Cost + Annual Holding Cost + Annual Backordering Cost
Annual Setup (Ordering) Cost = (D/Q) . S
Annual Holding Cost = (Average Inventory Level) . H

**20. **Backorder Inventory Model

**21. **Backorder Inventory Model By using the graphical ratios, we know that:
T1 / T = (Q ? b) / Q Therefore, if we replace T1/T in the above equation we get
Average Inventory Level = (Q ? b)2 / 2Q

**22. **Backorder Inventory Model

**23. **Backorder Inventory Model By using the graphical ratios, we know that:
T2 / T = b / Q Therefore, if we replace T2/T in the above equation we get
Average Backordering = b2 / 2Q and

**24. **Backorder Inventory Model

**25. **Backorder Inventory Model We find optimum order quantity (Q*) and optimum backordering quantity (b*) by taking the derivatives of dTC/dQ = 0 and dTC / db = 0 and then putting the values in their places.

**26. **Backorder Inventory Model

**28. **Quantity Discount Model A quantity discount is simply a reduced price (P) for an item when it is purchased in LARGER quantities.
A typical quantity discount schedule is as follows:

**29. **Quantity Discount Model

**30. **Quantity Discount Model Since the unit cost for the Third discount is the lowest, We might be tempted to order 2000 or more units.
However, this quantity might not be the one that minimizes the Total Cost.
Remember that, As the quantity goes up, the holding cost increases.

**31. **Quantity Discount Model Here, there is a trade off between reduced product price (P) and increased holding cost (H).
Total Cost = Setup Cost + Holding Cost + Product Price (Cost)
Total Cost = DS / Q + QH / 2 + PD where P is the price per unit

**32. **Quantity Discount Model To determine the minimum Total Cost, we perform the following process which includes 4 steps:

**33. **Quantity Discount Model Step 1: Assume that
I: is a percentage value, and
I . P represents the holding cost as a percentage of price per unit (P).

**34. **Quantity Discount Model For each discount alternative, calculate a value of Q* = [2DS / IP]1/2
Here, instead of using a value of H, the holding cost is equal to I . P
That is, If the item is expensive (such as a Class A Item), Its holding cost will be higher.

**35. **Quantity Discount Model Since the price of item (P) is a factor in Annual Holding Cost, we can no longer assume that the holding cost is constant (such as H) when price changes.

**36. **Quantity Discount Model Step 2: For any discount alternative,
If the calculated optimum order quantity (Q*) is too low to qualify for the discount range,
Then, Adjust the order quantity upward to the lowest quantity that will qualify for the particular discount alternative.

**37. **Quantity Discount Model Step 3: Using the total cost (TC) equation above, compute a total cost for every order quantity (Q). Use the adjusted Q values.

**38. **Quantity Discount Model Step 4: Select the discount alternative which has the minimum Total Cost (TC).

**39. **Example Consider the quantity discount schedule given in the beginning (above).
Assume that the Ordering (Setup) Cost (S) is $49 per each order.
Annual Demand (D) is 5000 units, and
Inventory carrying charge is a percentage (I=0.20) of product cost (P).

**40. **Example Question: What order quantity will minimize the total inventory cost.
Answer:
Step 1: Compute Q* for every discount range.

**41. **Example

**42. **Example Step 2: Adjust values of Q* that are below allowable discount ranges.
- For Q1, allowable range is 0-999. Since Q1* = 700 is between 0 and 999, It does not have to be adjusted.

**43. **Example - For Q2, allowable range is 1000-1999. Since Q2* = 714 is not in the allowed range, we adjust it to the lowest allowable value, That is Q2* = 1000.
- For Q3, allowable range is 2000-. Since Q3* = 718 is not in the allowed range, we adjust it to the lowest allowable value, That is Q3* = 2000.

**44. **Example Step 3: Compute total cost for each of the order quantities (Q*)

**45. **Example

**46. **Example Step 4: An Order quantity of 1000 units will minimize the total cost.
However, if the third discount cost is lowered to $4.65, selecting This discount alternative (2000 units) would be the optimum solution.

**47. **Probabilistic Models So far we assumed that demand is constant and uniform.
However, In Probabilistic models, demand is specified as a probability distribution.
Uncertain demand raises the possibility of a stock out (or shortage). (Why?)

**48. **Probabilistic Models One method of reducing stock outs is to hold extra inventory (called Safety Stock).
In this case, we change the ROP formula to include that safety stock (ss).

**49. **Probabilistic Models ROP = d . L
d = daily demand, and
L = Order Lead Time
Now it will be as follows:
ROP = d . L + (ss)
where (ss) is the safety stock

**50. **Example AMP Ltd. company determined its ROP = 50 units.
Its holding cost (H) is $5 per unit per year.
Its stock out cost (B) is $40 per unit.
Probability of stock out is based on the following probability distribution:

**51. **Example

**52. **Example Question: Find the Level of Safety Stock (ss) that minimizes the total additional holding cost and Stock out costs (annually).
Stock out cost is an expected cost; That is:

**53. **Example

**54. **Example Possible stock outs per year is actually the number of orders per year (D/Q).
Since it is not known (or not given) assume that it is 6 times / year.
For zero safety stock, there is no additional holding cost for extra (safety) stock.
But there are stock out costs for two levels:

**55. **Example 1) If demand is 60 units at the ROP, Then a shortage of 10 units will occur.
(Because, ROP is 50 units)
2) If demand is 70 units at the ROP, Then a shortage of 20 units will occur.

**56. **Example

**57. **Example The safety stock with the lowest total cost is (ss = 20) units.
Therefore, ROP = 50 + 20 = 70 units.

**58. **Finding A Safety Stock Level Managers may want to limit the possibility of stock out only to a small percentage, say 5%.

**59. **Finding A Safety Stock Level If demand level is assumed to be a normal distribution,
By using mean and standard deviation of the normal distribution,
We can determine a safety stock that is necessary for %95 service level.

**60. **Example The SAC company carries an inventory item that has a normally distributed demand.
The mean demand is (?=350) units and standard deviation is (?=10).

**61. **Example

**62. **Example

**63. **Example We use the properties of a standardized normal curve to get a z value that corresponds to the.95 of the curve.
Using a standard Normal table we find z = 1.65 for 95% confidence.

**64. **Example

**65. **Example