1. Inventory Control Models
Chapter 4
Inventory
Control Models

2. Learning Objectives Students will be able to:
Understand the importance of inventory control and ABC analysis.
Use the economic order quantity (EOQ) to determine how much to order.
Compute the reorder point (ROP) in determining when to order more inventory.
Solve inventory problems that allow quantity discounts or non-instantaneous receipt.
Understand the use of safety stock.
Describe the use of material requirements planning in solving dependent-demand inventory problems.
Discuss just-in-time inventory concepts to reduce inventory levels and costs.
Discuss enterprise resource planning systems.

3. Chapter Outline 6.7 Quantity Discount Models. 6.8 Use of Safety Stock.
6.1 Introduction. 6.2 Importance of Inventory Control. 6.3 Inventory Decisions. 6.4 Economic Order Quantity (EOQ): Determining How Much to Order. 6.5 Reorder Point: Determining When to Order. 6.6 EOQ without the Instantaneous Receipt Assumption.
6.7 Quantity Discount Models.
6.8 Use of Safety Stock.

4. Inventory as an Important Asset
1. INTRODUCTION
Inventory as an Important Asset
Inventory can be the most expensive and the most important asset for an organization.
Other Assets
60%
Inventory
40%
Inventory as a
percentage of total assets

5. Inventory Planning and Control
(Fig. 6.1)
Inventory Planning and Control
Planning on What Inventory to Stock and How to Acquire It
Forecasting Parts/Product Demand
Controlling Inventory Levels
Feedback Measurements to Revise Plans and Forecasts

6. The Inventory Process Suppliers Customers Raw Materials Finished Goods
Inventory Storage
Work in
Process
Fabrication and
Assembly
Inventory Processing

7. 2. Importance of Inventory Control
Five Functions of Inventory:
The decoupling function (inventory can act as a buffer).
Storing resources.
Responding to irregular supply and demand.
Taking advantage of quantity discounts.
Avoiding stockouts and shortages, and then maintain goodwill.

8. Five uses of inventory:
1. Decouple manufacturing processes.
Inventory is used as a buffer between stages in a manufacturing process.
This reduces delays and improves efficiency.
2. Storing resources.
Seasonal products may be stored to satisfy off-season demand.
Materials can be stored as raw materials, work-in-process, or finished goods.
Labor can be stored as a component of partially completed subassemblies.
3. Compensate for irregular supply and demand.
Demand and supply may not be constant over time.
Inventory can be used to buffer the variability.

9. Five uses of inventory:
4. Take advantage of quantity discounts.
Lower prices may be available for larger orders.
Extra costs associated with holding more inventory must be balanced against lower purchase price.
5. Avoid stockouts and shortages.
Stockouts may result in lost sales.
Dissatisfied customers may choose to buy from another supplier.

10. Overall goal is to minimize
3. Inventory Decisions
Two fundamental decisions in controlling inventory:
How much to order,
When to order.
Overall goal is to minimize
total inventory cost.

11. Inventory Costs Cost of the items “purchase cost or material cost ),
Cost of ordering (set-up or placing an order), “ordering (setup) cost ”,
Cost of carrying, or holding inventory, “holding cost or carrying cost ”,
Cost of stockouts, “stockout cost or shortage cost”.

12. Ordering (Set-up) Costs:
Developing and sending purchase orders,
Processing and inspecting incoming inventory,
Inventory inquiries,
Utilities, phone bills, etc., for the purchasing department,
Salaries/wages for purchasing department employees,
Supplies (e.g., forms and paper) for the purchasing department.

13. Carrying Costs: Cost of capital, Taxes, Insurance, Spoilage (damage),
Theft,
Obsolescence,
Salaries/wages for warehouse employees,
Utilities/building costs for the warehouse,
Supplies (e.g., forms, paper) for the warehouse.

14. Inventory Cost Factors
Ordering costs are generally independent of order quantity.
Many involve personnel time.
The amount of work is the same no matter the size of the order.
Carrying costs generally varies with the amount of inventory, or the order size.
The labor, space, and other costs increase as the order size increases.
The actual cost of items purchased can vary if there are quantity discounts available.

15. EOQ - Basic Assumptions:
4. Economic Order Quantity (EOQ):
Determining How Much to Order
EOQ - Basic Assumptions:
Demand is known and constant.
Lead time is known and constant.
Receipt of inventory is instantaneous.
Purchase cost is constant (Quantity discounts are not possible).
The only variable costs are: ordering cost, and holding cost.
Orders are placed so that stockouts or shortages are avoided completely .

16. Inventory Usage Over Time
Fig. 6.2: Inventory Usage Over Time
Receipt of inventory is instantaneous

17. Inventory Costs in the EOQ Situation
Average inventory level = (6.1)
Variables:
Q = number of pieces to order.
EOQ = Q* = optimal number of pieces to order,
D = annual demand in units for the inventory items,
Co = ordering cost of each order,
Ch = holding or carrying cost per unit per year.

18. Average inventory level
DAY
BEGINNING
ENDING
AVERAGE
April 1 (order received) 10 8 9
April 2 6 7
April 3 4 5
April 4 2 3
April 5 1
Maximum level April 1 = 10 units Total of daily averages = = 25 Number of days = 5 Average inventory level = 25/5 = 5 units = 10/2

19. Total Cost as a Function of Order Quantity
Fig. 6.3
Total Cost as a Function of Order Quantity
Minimum Total Cost
Optimal Order Quantity
Curve of Total Cost of Carrying and Ordering
Carrying Cost Curve
Ordering Cost Curve
Cost
Order Quantity

20. Finding the EOQ Set the ordering cost equal to the carrying cost.
Develop an expression for the ordering cost.
Develop an expression for the carrying cost.
Set the ordering cost equal to the carrying cost.
Solve this equation for the optimal order quantity, Q*.

21. Annual holding or carrying cost =
Annual ordering cost =
(number of orders placed per year) × (ordering cost per order)
annual demand
= × (ordering cost per order)
number of units in each order
Annual holding or carrying cost =
= (average inventory) × (carrying cost per unit per year)
order quantity
= × (carrying cost per unit per year) 2

22. Setting the Equations Equal to Solve for Q*
Annual ordering cost = Annual holding cost: h C 2 Q o D =
Q 2 h C 2 o D = Q* = h C 2 o D
EOQ =

23. Economic Order Quantity (EOQ) Model
Average inventory level = (6.1) o C Q D
Annual ordering cost =
(6-2)
Annual holding cost h C 2 Q =
(6-3) Q* = h C 2 o D
EOQ =
Economical Order Quantity
(6-4)
Total inventory cost =
(6-5)

24. Sumco Pump Company Example
Sumco, a company that sells pump housings to other manufacturers, would like to reduce its inventory cost by determining the optimal number of pump housings to obtain per order. The annual demand is 1,000 units, the ordering cost is $10 per order, and the average carrying cost per unit per year is $0.50. Using these figures, if the EOQ assumptions are met, we can calculate the optimal number of units per order:

25. Sumco Pump Company Example
Sumco, a company that sells pump housings to other manufacturers, would like to reduce its inventory cost by determining the optimal number of pump housings to obtain per order (Q* ?). The annual demand is 1,000 units (D), the ordering cost is $10 per order (Co), and the average carrying cost per unit per year is $0.50 (Ch).

26. the optimal number of units per order:Q* = h C 2 o D
EOQ =
= ,000 (10) = 40,000 = 200 units
- Total inventory costs =
= 1, (1) (0.50) = = $100
- The number of orders per year = (D/Q) = 1,000 / 200 = 5
- The average inventory = Q/2 = 200/ 2 = 100

27. Lab Exercise # 3
1. Solve Sumco Pump Company Example using Excel and QM for Windows.
(page ……).
To be Continued.

28. Solution Using Excel = =Q* =Q*/2 =D/Q* =Average Inventory*Ch
=Number of Orders*Setup (order) cost
=Unit price*Demand
=Holding costs+Setup costs+Unit costs
Total
Solution Using Excel h C 2 o D =

29. Total

31. Solution Using QM for Windows

35. Enter the Input Parameters, then Click Solve

38. Purchase Cost of Inventory Items
Let: C = the purchase cost per unit .
The average dollar level = (6-6)
Typically, carrying cost, Ch, is stated in:
$ cost per unit per year.
Sometimes, it is expresses as percentage of the unit cost.
Let I = The annual inventory holding cost as a percent of unit cost, then:
Ch = IC

39. EOQ = Q Per Unit Carrying Cost: * 2DC = Q C Denominator Change
Percentage Carrying Cost: Q * = IC
2DC o
(6-7)

40. Sensitivity Analysis with the EOQ Model
Determining the effects of changes in input values on the EOQ is called sensitivity analysis.
From the equation for Q* : the EOQ changes by the square root of a change in any of the inputs D, C○, Ch) . * Q = h C
2DC o

41. Sumco Pump Company Example (Revisited)
Consider the Sumco Pump Company example. What will happen to the EOQ if we increased C○ from $10 to $40. Q* = h C 2 o D
EOQ =
= ,000 (40) = 400 units

42. Inputs and Outputs of the EOQ Model
Input Values
Output Values
Annual Demand (D)
Ordering Cost (Co)
Carrying Cost (Ch)
Lead Time (L)
Demand Per Day (d)
Economic Order
Quantity (EOQ)
Reorder Point
(ROP)
# of Orders = (D/Q)
Avg. Inventory = (Q/2)
Cycle Time = Time between orders
= 360/#of orders
= 360 × (Q/D)
EOQ
Models

43. 5. Reorder Point (ROP): Determining When to Order
ROP = (Demand per day) x (Lead time for a new order, in days) ROP = d x L
……….(6.8)
Inventory
Level
(Units)
Q *
ROP
Slope = Units/Day = d
Lead Time (Days) = L
Time (Days)

44. 6. EOQ without the Instantaneous Receipt Assumption
Production Run Model:
Inventory Level
There is No Production During This Part of the Cycle
Maximum Inventory
Level
Time
Production Portion of Cycle
Inventory Control and the Production Process (Figure 6.5) t

45. Annual Holding (Carrying) Cost for Production Run Model
Let:
Q = number of pieces per production run, or (or per order).
t = length of production run in days, or
(length of time for receiving the inventory order).
Cs = Setup cost (Annual Ordering Cost),
Ch = Holding (or carrying cost) per unit per year.
p = daily production rate, or (daily receiving rate).
d = daily demand rate.

46. Average annual inventory = ……(6.9)
Q = number of pieces produced,
p = daily production rate,
d = daily demand rate
t = length of production run in days,
Ch carrying cost per unit per year
The maximum inventory =
= (total production during the production run) –
(total used during the production run)
= (pt ) – (dt ) = t (p – d)
= (p – d ), The maximum inventory =
Average annual inventory = one-half of the maximum inventory
Average annual inventory = ……(6.9)
Annual holding cost = (average annual inventory) × (holding cost)
Annual holding cost =
(6-10)

47. Annual Setup Cost, or
Annual Ordering Cost
Both of these are independent of the size of the production run (size of the order).
This cost = the number of production runs (number of orders) times the setup cost (ordering cost).
Annual Setup Cost =
Annual Ordering Cost = s C Q D
(6-11) o C Q D
(6-12)

48. Determining the Optimal Production Quantity
Costs are minimized when the setup cost (ordering cost) equals the holding cost. s C Q D = = p d 1 C
2 D C Q h s * _
(6-13)
If the case involves the receipt of inventory over a period of time, use the same model but replace the setup cost Cs with the ordering cost Co .

49. Production Run Model 2 D C Q = d C 1 p D Annual Setup Cost = C Q _
Annual holding cost =
Annual Setup Cost = s C Q D = p d 1 C
2 D C Q h s * _
Optimal Production Quantity

50. Brown Manufacturing Example
Brown Manufacturing produces commercial refrigeration units in batches. The firm’s estimated demand for the year is 10,000 units. It costs about $100 to set up the manufacturing process, and the carrying cost is about 50 cents per unit per year. When the production process has been set up, 80 refrigeration units can be manufactured daily. The demand during the production period has traditionally been 60 units each day. Brown operates its refrigeration unit production area 167 days per year. How many refrigeration units should Brown Manufacturing produce in each batch? How long should the production part of the cycle shown in Figure 6.5 last?

51. Brown Manufacturing Example
Brown Manufacturing produces commercial refrigeration units in batches. The firm’s estimated demand for the year is 10,000 units. It costs about $100 to set up the manufacturing process, and the carrying cost is about 50 cents per unit per year. When the production process has been set up, 80 refrigeration units can be manufactured daily. The demand during the production period has traditionally been 60 units each day. Brown operates its refrigeration unit production area 167 days per year. How many refrigeration units should Brown Manufacturing produce in each batch? How long should the production part of the cycle shown in Figure 6.5 last?

52. Brown Manufacturing Example
Brown Manufacturing produces commercial refrigeration units in batches. The firm’s estimated demand for the year is 10,000 units. It costs about $100 to set up the manufacturing process, and the carrying cost is about 50 cents per unit per year. When the production process has been set up, 80 refrigeration units can be manufactured daily. The demand during the production period has traditionally been 60 units each day. Brown operates its refrigeration unit production area 167 days per year. How many refrigeration units should Brown Manufacturing produce in each batch? How long should the production part of the cycle shown in Figure 6.5 last?
Given Data:
Annual demand (D) = 10,000 units.
Setup cost (Cs) = $100.
Carrying cost (Ch) = $0.50 per unit per year.
Production rate (p) = 80 units daily.
Demand rate (d) = 60 units daily.
Refrigeration operational days = 167 days per year.
Optimal Production Quantity Q* ?
t = Q*/p ?

53. Brown Manufacturing Example
Questions
How many refrigeration units should Brown produce in each batch?
i.e., What is Q* ?
How long should the production cycle last ?
i.e., What is t = Q*/p ?

54. What is Q* ? è ö ø ö è æ æ = p d 1 C 2 D C Q ø = 1 (0.5)_ ö ø ø ö è æ = 1
(0.5)
2(10,000)(100) Q * _ 60 80 =
4,000 units

55. Production runs will last 50 days and produce 4,000 units.
How long should the production cycle lasts ? What is t (Q/p)? Q * =
4,000 units.
p = 80 units per day.
t = Q*/p = 4,000/80 = 50 days.
Production runs will last 50 days and produce 4,000 units.

56. Lab Exercise # 3 (Cont’d.)
2. a) Solve the Brown Manufacturing Example using Excel QM and QM for Windows. b) Draw a curve (in scale) representing the inventory control and the production process (as shown in Figure 6.5).
To be Continued.

58. =SQRT((2*B7*B8)/(B9*(1-B11/B10)))p d 1 C
2 D C Q h s * _
=SQRT((2*B7*B8)/(B9*(1-B11/B10)))
=B15*(1-(B11/B10)) =
=B16/ = ½ Maximum inventory
=B7/B = D/Q*
=B17*B = Average inventory * Holding cost
=B18*B = Number of Setups * Setup cost
=B20+B = Total Holding cost + Total Setup cost

64. Enter data and Click Solve

67. 7. Quantity Discount Models
Total inventory cost =
Purchase cost + Ordering cost + Holding cost
TC = DC + (D/Q)Co + (Q/2)Ch (6-14)
Where:
D = annual demand in units.
C = cost per unit.
Co = ordering cost for each order.
Ch = holding cost per unit per year.
Let I = holding cost as a percentage of the unit cost (C).
Ch = IC, IC is convenient to be used in place of Ch in decision-making.

68. Total Cost Curve for each of the Quantity Discount Model
EOQ = Q*
Figure 6.6
The lowest cost quantity for the curve

69. Quantity Discount Models
1. For each discount price (C), Compute EOQ :
2. If EOQ < Minimum for discount, adjust the quantity to Q = Minimum for discount.
3. For each EOQ, compute total cost:
TC = DC + D/Q(Co) + Q/2(Ch)
4. Choose the lowest cost quantity from all levels. Q * = IC
2DC o Q* = h C 2 o D
EOQ =

70. Brass Department Store Example
Brass Department Store stocks toy race cars. Recently, the store was given a quantity discount schedule for the cars; this quantity discount schedule is shown in Table 6.3. Thus, the normal cost for the toy race cars is $5. For orders between 1,000 and 1,999 units, the unit cost is $4.80, and for orders of 2,000 or more units, the unit cost is $4.75. Furthermore, the ordering cost is $49 per order, the annual demand is 5,000 race cars, and the inventory carrying charge as a percentage of cost, I, is 20% or 0.2. What order quantity will minimize the total inventory cost?

71. Brass Department Store Example
Brass Department Store stocks toy race cars. Recently, the store was given a quantity discount schedule for the cars; this quantity discount schedule is shown in Table 6.3. Thus, the normal cost for the toy race cars is $5. For orders between 1,000 and 1,999 units, the unit cost is $4.80, and for orders of 2,000 or more units, the unit cost is $4.75. Furthermore, the ordering cost is $49 per order, the annual demand is 5,000 race cars, and the inventory carrying charge as a percentage of cost, I, is 20% or 0.2. What order quantity will minimize the total inventory cost?

72. Brass Department Store Example
Quantity Discount Schedule
Table 6.3
Material cost:
Total material cost is affected by the Discount (%).
Unit cost = first $5.00, then $4.80, and finally $4.75

73. The ordering cost Co = $49 per order.
Brass Department Store Example (Cont’d)
The ordering cost Co = $49 per order.
The annual demand D = 5000 race cars.
The holding cost as a percentage of purchase cost (I) = 20% or
What order quantity will minimize the total inventory cost?

74. Solution Steps: The first step is to compute EOQ for every discount:
EOQ1 = = 700 cars/order
EOQ2 = = 714 cars/order
EOQ3 = = 718 cars/order
EOQ = Q * = IC
2DC o

75. EOQ1 = 700 is between 0 and 999, it does not have to be adjusted.
2. The second step is to adjust those quantities that are below the allowable discount range:
EOQ1 = 700 is between 0 and 999, it does not have to be adjusted.
EOQ2 = 714 is below the allowable range of 1000 to 1999, and therefore, it must be adjusted to 1000 units.
The same is true for EOQ3 = 718; it must be adjusted to 2000 units.
So, Q1 = 700,
Q2 = 1000 (adjusted),
Q3 = 2000 (adjusted).

76. TC = DC + D/Q (Co) + Q / 2 (Ch)
3. The third step is to compute a total cost for each of the order quantities:
TC = DC + D/Q (Co) + Q / 2 (Ch)
D=5000, Co= $49, Ch = IC =0.2C
Total
Cost
Annual
Holding
Annual Ordering
Annual Material DC
Order
Quantity
(Q)
Unit
Price
( C )
Discount
Number
25700
350
25000
700 $5 1
24725
480
245
24000
1000
$4.8 2
24822
950
122
23750
2000
$4.75 3
Table 6.4
4. The fourth step is to select that order quantity with the lowest total cost:
Q* = 1000 units with total cost = $24,725 .

77. Lab Exercise # 3 (Cont’d)
3. Solve the Brass Department Store using Excel QM and QM for Windows.

78. IF(logical_test,[value_if_true],[value_if_false])
Syntax

86. 8. Use of Safety Stock ROP = d × L + SS ….(6-15)
Safety stock is extra stock on hand to avoid stockouts.
Stockouts occur when there are uncertainties with:
Demand,
Lead time.
ROP is adjusted to implement safety stock policy:
ROP = d × L + SS ….(6-15)
d = average daily demand,
L = average lead time,
SS = safety stock.

87. Use of Safety Stock Fig. 6.7 Time Stockout Time Safety Stock, SS
Inventory on
Hand
Time
Inventory on
Hand
Stockout
is avoided
Time
Safety
Stock, SS
0 Units
Stockout

88. Safety Stock with Unknown Stockout Costs
Set service level; use normal distribution for Demand Lead time.
When stockout costs are not quantifiable or not applicable:
Use a service level to determine safety stock level.
Service level is the percent of time an item is not out of stock.
Service Level = 1 – P(Stockout)
Or:
P(Stockout) = 1 – Service Level

89. Hinsdale Company Example:
The Hinsdale Company carries a variety of electronic inventory items, and these are typically identified by SKU. One particular item, SKU A3378, has a demand that is normally distributed during the lead time, with a mean of 350 units and a standard deviation of 10. Hinsdale wants to follow a policy that results in stockouts occurring only 5% of the time on any order. How much safety stock should be maintained and what is the reorder point? Figure 6.8 helps visualize this example.

90. Hinsdale Company Example:
The Hinsdale Company carries a variety of electronic inventory items, and these are typically identified by SKU. One particular item, SKU A3378, has a demand that is normally distributed during the lead time, with a mean of 350 units and a standard deviation of 10. Hinsdale wants to follow a policy that results in stockouts occurring only 5% of the time on any order. How much safety stock should be maintained and what is the reorder point? Figure 6.8 helps visualize this example.
Lead time demand ~N(350, 10)
P (Stockout) = 5%

91. Hinsdale Company Example:
Lead time demand ~N(350, 10)
Where: Average (m) = 350, Standard deviation (s) = 10.
Desired Policy:
P (Stockout) = 5%
Therefore, service level = 95%
Visualization of Desired Inventory Policy:
μ = Mean demand = 350
σ = Standard deviation = 10
X = Mean demand + Safety stock
Figure 6.8

92. Hinsdale Company example:
X = Mean demand m + Safety Stock (SS) Z = X – m = SS = Zs
SS = X – μ
X-m s
SS = Z σ
Z is obtained from the Normal distribution tables.

93. Hinsdale Company example:
Find Z using a Normal table, like in Appendix A:
Z = 1.65 for a 5% right tail ( Search for an area of inside the table).
SS = Zσ = 1.65 (10) = 16.5 units,
or 17 units.
ROP = Average demand during lead time + SS
ROP = μ +SS = = 367 units.

94. 0.95
Z = 1.65

95. Service Level versus Holding Costs
Hinsdale Company Example:
If the carrying cost Ch = $2 per unit per year. The carrying cost will vary for service levels that range from 90% 91%, 92%,…, 99.99%.
Table 6.5 :
Service Level (%)
Z value from Normal Curve Table
Safety Stock (Units)
Holding Cost of the Safety Stock 90
1.28
12.8
25.6 91
1.34
13.4
26.8 92
1.41
14.1
28.2 93
1.48
14.8
29.6 94
1.55
15.5 31 95
1.65
16.5 33 96
1.75
17.5 35 97
1.88
18.8
37.6 98
2.05
20.5 41 99
2.33
23.3
46.6
99.99
3.75
37.2
74.4

96. Service Level versus Carrying Costs
The following curve depicts the tradeoff between carrying costs of safety stocks and service level for the previous example.
Such dramatic tradeoffs exist for all similar problems

97. Service Level versus Annual Carrying Costs
Holding cost of safety Stock ($)
Figure 6.9

98. Total Annual Holding Cost
holding cost of regular inventory + Holding cost of safety stock
THC = Q/2 Ch + (SS) Ch …. (6.20)
Where:
THC = total annual holding cost,
Q = order quantity,
Ch = holding cost per unit per year,
SS = safety stock.