1. MNG221- Management Science –
Inventory Management
MNG221- Management Science –

2. Lecture Outline Types of inventory Reasons for holding inventory
Stock costs
Objective of inventory management
Pareto analysis
Deterministic and stochastic models

3. Inventory Management Stock may be classified into: Raw materials
Work-in-progress
Finished goods
Resources, Labor, Cash
The classification depends on the nature of the firm.

4. Inventory Management
The main purpose of inventory is simply to meet customer demand.
It often represents a significant cost to a business firm, (including insurance, obsolescence, depreciation, interest, opportunity costs, storage costs, etc.)
Therefore inventory related costs can be controlled, through the management of inventory levels.

5. Elements of Inventory Management

6. Elements of Inventory Management
Inventory is defined as a stock of items kept on hand by an organization to use to meet customer demand.
The Role of Inventory
The main reasons for holding inventory are:
To satisfy demand immediately
To meet seasonal or cyclical demand

7. Elements of Inventory Management
The Role of Inventory
To allow for unimpeded production and provide independence between operations.
To take advantage of bulk purchasing price discounts.
To absorb seasonal fluctuations.
A necessary part of the production process.

8. Elements of Inventory Management
The Role of Inventory (continued)
Inventory may also accumulate because of poor control methods, obsolesce and suboptimal decisions.

9. Elements of Inventory Management
Demand
A crucial component and the basic starting point for the management of inventory is customer demand, because it exists for the purpose of meeting the demand of customers.
Customers can be Internal (machine operator) or External (Individual purchasing goods from stores)

10. Elements of Inventory Management
Demand (Continued)
An essential determinant of effective inventory management is an accurate forecast of demand.
The demand for items in inventory is classified as dependent or independent
Dependent Demand items are used internally to produce a final product
Independent Demand items are final products demanded by an external customer.

11. Elements of Inventory Management
Inventory Costs
There are three basic costs associated with inventory:
Carrying Costs - are the costs of holding items in storage.
Ordering Costs - are the costs associated with replenishing the stock of inventory being held.

12. Elements of Inventory Management
Inventory Costs
Shortage costs - also referred to as stockout costs, occur when customer demand cannot be met because of insufficient inventory on hand.

13. Elements of Inventory Management
Inventory Costs
The objective of inventory management is to employ an inventory control system that will indicate how much should be ordered and when orders should take place to minimize the sum of the above three inventory costs

14. Inventory Control Systems
Inventory Management
Inventory Control Systems

15. Inventory Control Systems
An Inventory System is a structure for controlling the level of inventory by determining how much to order (the level of replenishment) and when to order.
There are two basic types of inventory systems: a continuous (or fixed order quantity) system and a periodic (or fixed time period) system.

16. Inventory Control Systems
The primary difference in the two systems is that in a:
Continuous system - an order for the same amount is placed whenever the inventory decreases to a certain level.
Periodic system - order is placed for a variable after an established passage of time.

17. Inventory Control Systems
Continuous Inventory System
In a continuous inventory system, (alternatively referred to as a perpetual system or a fixed order quantity system) a constant amount is ordered when inventory declines to a predetermined level, referred to as the reorder point.
This fixed order quantity is called the economic order quantity

18. Inventory Control Systems
Continuous Inventory System
The inventory level is closely and continuously monitored so that management always knows the inventory status.
However, the cost of maintaining a continual record of the amount of inventory on hand can also be a disadvantage of this type of system.

19. Inventory Control Systems
Periodic Inventory System
In a periodic inventory system, (also referred to as a fixed time period system or periodic review system) an order is placed for a variable amount after a fixed passage of time.

20. Inventory Control Systems
Periodic Inventory System
The inventory level is not monitored at all during the time interval between orders.
It has the advantage of requiring little or no record keeping
It has the disadvantage of less direct control

21. Economic Order Quantity Models Basic Model
Inventory Management
Economic Order Quantity Models Basic Model

22. Economic Order Quantity Models
The most widely used and traditional means for determining how much to order in a continuous system is the Economic Order Quantity (EOQ) model, also referred to as the Economic Lot Size Model.
The function of the EOQ model is to determine the optimal order size that minimizes total inventory costs.

23. Economic Order Quantity Models
The Basic EOQ Model
It is essentially a single formula for determining the optimal order size that minimizes the sum of carrying costs and ordering costs.

24. Economic Order Quantity Models
The Basic EOQ Model
Assumptions
Demand is known with certainty and is relatively constant over time.
No shortages are allowed.
Lead time for the receipt of orders is constant.
The order quantity is received all at once.

25. Economic Order Quantity Models
The Basic EOQ Model
The Inventory Order Cycle

26. Economic Order Quantity Models
The Basic EOQ Model
Q is the point at which ordering and carrying costs react inversely to each other in response to an increase in the order size.
R is the point at which a new order is placed with enough lead time for the reordering of stock.

27. Economic Order Quantity Models
The Basic EOQ Model – Carrying Costs
Carrying cost is usually expressed on a per-unit basis for some period of time on an annual basis (i.e., per year), and sometimes as a percentage of average inventory.
Average Inventory = Q or ∑Q points over period, t
number of points

28. Economic Order Quantity Models
The Basic EOQ Model – Carrying Costs

29. Economic Order Quantity Models
The Basic EOQ Model – Carrying Costs
Thus,
Carrying cost is
Ordering Costs is
Total Inventory Cost is
The EOQ cost model

30. Economic Order Quantity Models
The Basic EOQ Model
The Optimal Value Of Q corresponds to the lowest point on the total cost curve or the point where the ordering cost curve intersects with the carrying cost curve.

31. Economic Order Quantity Models
The Basic EOQ Model
Thus The Optimal Value Of Q by equating the two cost functions and solving for Q, as follows:

32. Economic Order Quantity Models
The Basic EOQ Model
Alternatively, the optimal value of Q can be determined by differentiating the total cost curve with respect to Q

33. Economic Order Quantity Models
The Basic EOQ Model
The total minimum cost

34. Economic Order Quantity Models
The Basic EOQ Model - Example
The I-75 Carpet Discount Store wants to determine the optimal order size and total inventory cost given an estimated annual demand of 10,000 yards of carpet, an annual carrying cost of $0.75 per yard, and an ordering cost of $150.
The store would also like to know the number of orders that will be made annually and the time between orders (i.e., the order cycle).

35. Economic Order Quantity Models
The Basic EOQ Model – Example
The model parameters as follows:

36. Economic Order Quantity Models
The Basic EOQ Model – Example
The optimal order size is computed as follows:

37. Economic Order Quantity Models
The Basic EOQ Model – Example
The total annual inventory cost is determined by substituting Qopt into the total cost formula, as follows:

38. Economic Order Quantity Models
The Basic EOQ Model – Example
The number of orders per year is computed as follows:

39. Economic Order Quantity Models
The Basic EOQ Model – Example
Given that the store is open 311 days annually (365 days minus 52 Sundays, plus Thanksgiving and Christmas), the order cycle is determined as follows:

40. Economic Order Quantity Models
The Basic EOQ Model
The optimal order quantity determined in general, is an approximate value, because it is based on estimates of carrying and ordering costs as well as uncertain demand.
This in practice it is acceptable to round off the Q values to the nearest whole number.
However, the EOQ model is robust; because Q is a square root, errors in the estimation of D, Cc, and Co are dampened.

41. Economic Order Quantity Models Non-instantaneous Model
Inventory Management
Assumptions
Demand is known with certainty and is relatively constant over time.
No shortages are allowed.
Lead time for the receipt of orders is constant.
The order quantity is received all at once.
Economic Order Quantity Models Non-instantaneous Model

42. Economic Order Quantity Models
Non-instantaneous Receipt Model
A variation of the basic EOQ model is achieved when the assumption that orders are received all at once is relaxed.
It is also referred to as the Gradual Usage, or Production Lot Size, model.
In this EOQ variation, the order quantity is received gradually over time and the inventory level is depleted at the same time it is being replenished.

43. Economic Order Quantity Models
Non-instantaneous Receipt Model
This is a situation most commonly found when the
Inventory user is also the producer
When orders are delivered gradually over time
When retailer and producer of a product are one and the same.

44. Economic Order Quantity Models
Non-instantaneous Receipt Model
The EOQ model with Non-instantaneous Order Receipt

45. Economic Order Quantity Models
Non-instantaneous Receipt Model
The ordering cost component of the basic EOQ model does not change.
However, the carrying cost component is not the same for this model variation because average inventory is different.
The maximum inventory level is not simply Q; it is an amount somewhat lower than Q,

46. Economic Order Quantity Models
Non-instantaneous Receipt Model
Unique parameters of this model:
p = daily rate at which the order is received over time, also known as the production rate
d = the daily rate at which inventory is demanded

47. Economic Order Quantity Models
Non-instantaneous Receipt Model
As such, the maximum amount of inventory that is on hand is computed as follows:

48. Economic Order Quantity Models
Non-instantaneous Receipt Model
Given the maximum inventory level, the average inventory level is determined by dividing this amount by 2, as follows:

49. Economic Order Quantity Models
Non-instantaneous Receipt Model
The total carrying cost, using this function for average inventory, is:

50. Economic Order Quantity Models
Non-instantaneous Receipt Model
Thus, the total annual inventory cost is determined according to the following formula:

51. Economic Order Quantity Models
Non-instantaneous Receipt Model
Therefore, to find optimal Qopt, we equate total carrying cost with total ordering cost:

52. Economic Order Quantity Models
Non-instantaneous Receipt Model Example
Assume that the I-75 Carpet Discount Store has its own manufacturing facility
further assume that the ordering cost, Co, is the cost of setting up the production process
Recall that Cc = $0.75 per yard and D = 10,000 yards per year.
The manufacturing facility operates 311 days and produces 150 yards of the carpet per day.

53. Economic Order Quantity Models
Non-instantaneous Receipt Model Example
Thus the parameters are:

54. Economic Order Quantity Models
Non-instantaneous Receipt Model Example
The optimal order size is determined as follows:

55. Economic Order Quantity Models
Non-instantaneous Receipt Model Example
This value is substituted into the following formula to determine total minimum annual inventory cost:

56. Economic Order Quantity Models
Non-instantaneous Receipt Model Example
The length of time to receive an order or production run is computed as follows:

57. Economic Order Quantity Models
Non-instantaneous Receipt Model Example
The number of orders per year is actually the number of production runs that will be made, computed as follows:

58. Economic Order Quantity Models
Non-instantaneous Receipt Model Example
Finally, the maximum inventory level is computed as follows:

59. Economic Order Quantity Models Shortages Model
Inventory Management
Economic Order Quantity Models Shortages Model

60. Economic Order Quantity Models
Shortages Model
The assumptions of our basic EOQ model is that shortages and back ordering are not allowed
The EOQ model with shortages relaxes the assumption that shortages cannot exist.
However, it will be assumed that all demand not met because of inventory shortage can be back ordered and delivered to the customer later.

61. Economic Order Quantity Models
Shortages Model
The EOQ model with Shortages

62. Economic Order Quantity Models
Shortages Model
Because back-ordered demand, or shortages, (S), are filled when inventory is replenished, the maximum inventory level does not reach Q, but instead a level equal to Q - S.
Therefore, the cost associated with shortages has an inverse relationship to carrying costs.
As the order size, Q, increases, the carrying cost increases and the shortage cost declines.

63. Economic Order Quantity Models
Shortages Model
Cost model with shortages

64. Economic Order Quantity Models
Shortages Model
The individual cost functions are provided as follows, where S equals the shortage level and Cs equals the annual per-unit cost of shortages:

65. Economic Order Quantity Models
Shortages Model
Combining these individual cost components results in the total inventory cost formula:

66. Economic Order Quantity Models
Shortages Model
The three cost component curves do not intersect at a common point, as was the case in the basic EOQ model.
As such, the only way to determine the optimal order size and the optimal shortage level, S, is to differentiate the total cost function with respect to Q and S.

67. Economic Order Quantity Models
Shortages Model
Set the two resulting equations equal to zero, and solve them simultaneously.
Doing so results in the following formulas for the optimal order quantity and shortage level:

68. Economic Order Quantity Models
Shortages Model Example
Assume that the I-75 Carpet Discount Store allows shortages and the shortage cost, Cs, is $2 per yard per year.
All other costs and demand remain the same

69. Economic Order Quantity Models
Shortages Model
Several additional parameters of the EOQ model with shortages can be computed for this example, as follows:
The time between orders, identified as t in is computed as follows:

70. Economic Order Quantity Models
Shortages Model
The time during which inventory is on hand, t1, and the time during which there is a shortage, t2, during each order cycle can be computed using the following formulas: