1. CHAPTER 11
Inventory Management

2. What is inventory?
An inventory is an idle stock of material used to facilitate production or to satisfy customer needs

3. Why do we need inventory?
Economies of scales in ordering or production - cycle stock
trade-off setup cost vs carrying cost
Smooth production when requirements have predictable variability - seasonal stock
trade-off production adjustment cost vs carrying cost

4. Why do we need inventory? (cont’d)
Provide immediate service when requirements are uncertain (unpredicatable variability) -safety stock
trade-off shortage cost versus carrying cost
Decoupling of stages in production/distribution systems -decoupling stock
trade-off interdependence between stages vs carrying cost

5. Why do we need inventory? (cont’d)
Production is not instantaneous, there is always material either being processed or in transit - pipeline stock
trade-off cost of speed vs carrying cost

6. Why carrying inventory
As you can see from the above list, there are economic reasons, technological reasons as well as management/operations reasons.

7. Why carrying inventory is costly?
Traditionally, carrying costs are attributed to:
Opportunity cost (financial cost)
Physical cost (storage/handling/insurance/theft/obsolescence/…)

8. Inventory Management
Independent Demand A
Dependent Demand
B(4)
C(2)
D(2)
E(1)
D(3)
F(2)
Independent demand is uncertain. Dependent demand is certain.

9. Types of Inventories Raw materials & purchased parts
Partially completed goods called work in progress
Finished-goods inventories
(manufacturing firms) or merchandise (retail stores)

10. Types of Inventories (Cont’d)
Replacement parts, tools, & supplies
Goods-in-transit to warehouses or customers

11. Inventory Counting Systems
Periodic System
Physical count of items made at periodic intervals
Perpetual Inventory System System that keeps track of removals from inventory continuously, thus monitoring current levels of each item

12. Inventory Counting Systems (Cont’d)
Two-Bin System - Two containers of inventory; reorder when the first is empty
Universal Bar Code - Bar code printed on a label that has information about the item to which it is attached

13. ABC Classification System
Classifying inventory according to some measure of importance and allocating control efforts accordingly.
A - very important
B - mod. important
C - least important A B C
High
Annual
$ volume
of items
Low
Few
Many
Number of Items

14. What are we tackling in inventory management?
All models are trying to answer the following questions for given informational, economic and technological characteristics of the operating environment:
How much to order (produce)?
When to order (produce)?
How often to review inventory?
Where to place/position inventory?

15. The inventory models
The quantitative inventory management models range from simple to very complicated ones. However, there are two simple models that capture the essential tradeoffs in inventory theory
The newsboy (newsvendor) model
The EOQ (Economic Ordering Quantity) model

16. The Newsboy Model
The newsboy model is a single-period stocking problem with uncertain demand: Choose stock level and then observe actual demand.
Tradeoff: overstock cost versus opportunity cost (lost of profit because of under stocking)

17. The Newsboy Model
Shortage cost (Cs) - the opportunity cost for lost of sales as well as the cost of losing customer goodwill.
Cs = revenue per unit - cost per unit
Excess cost (Ce) - the over-stocking cost (the cost per item not being able to sell)
Ce = Cost per unit - salvage value per unit

18. The Newsboy Model
In this model, we have to consider two factors (decision variables): the supply (X), also know as the stock level, and the demand (Y). The supply is a controllable variable in this case and the demand is not in our control. We need to determine the quantity to order so that long-run expected cost (excess and shortage) is minimized.

19. The Newsboy Model
Let X = n and P(n) = P(Y>n). The question is: Based on what should we stock this n-th unit?
The opportunity cost (expected loss of profit)for this n-th unit is P(n) Cs
The expected cost for not being able to sell this n-th unit (expected loss)
(1-P(n))Ce

20. The Newsboy model Stock the n-th unit if P(n)Cs > (1-P(n))Ce
Do not stock if P(n)Cs < (1-P(n))Ce

21. The Newsboy model The equilibrium point occurs at Solving the equation
P(n)Cs = (1-P(n)) Ce
Solving the equation
P(n) = Ce/(Cs + Ce)

22. Newsboy model
Service level is the probability that demand will not exceed the stocking level and is the key to determine the optimal stocking level.
In our notation, it is the probability that YX(=n), which is given by
P(Yn) = 1 - P(n) = Cs/(Cs + Ce)

23. Example (p.564)
Demand for long-stemmed red roses at a small flower shop can be approximated using a Poisson distribution that has a mean of four dozen per day. Profit on the roses is $3 per dozen. Leftover are marked down and sold the next day at a loss of $2 per dozen. Assume that all marked down flowers are sold. What is the optimal stocking level?

24. Example (solutions) Cs = $3, Ce = $2 Opportunity cost is P(n)Cs
Cumulative frequencies for Poisson distribution, mean = 4
Cs = $3, Ce = $2
Opportunity cost is P(n)Cs
Expected loss for over-stocking (1 - P(n)) Ce
P(Y  n) = Cs/(Cs + Ce) = 3/((3+2) = 0.6
From the table, it is between 3 and 4, round up give you optimal stock of 4 dozen.

25. Profile of Inventory Level Over Time
The Inventory Cycle
Profile of Inventory Level Over Time Q
Usage
rate
Quantity
on hand
Reorder
point
Time
Receive
order
Place
order
Receive
order
Place
order
Receive
order
Lead time

26. How to estimate the inventory costs?

27. Estimating the inventory costs
Q = quantity to order in each cycle
S = fixed cost or set up cost for each order
D = demand rate or demand per unit time
H = holding cost per unit inventory per unit time
c = unit cost of the commodity
TC = total holding cost per unit time

28. Estimating the inventory cost
The cost per order = S + cQ
The length of order cycle = Q/D unit time
The average inventory level = Q/2
Thus the inventory carrying cost
= (Q/2) H (Q/D) = HQ2/(2D)

29. Estimating the inventory cost
The total cost per order cycle
= S + cQ + HQ2/(2D)
Thus the total cost per unit time is dividing the above expression by Q/D, I.e.,
TC = {S + cQ + HQ2/(2D)} {D/Q}
= DS/Q + cD + HQ/2

30. Total Cost Annual carrying cost ordering Total cost = + Q 2 H D S TC =

31. Deriving the EOQ
Using calculus, we take the derivative of the total cost function and set the derivative (slope) equal to zero and solve for Q.

32. Cost Minimization Goal
The Total-Cost Curve is U-Shaped
Ordering Costs QO
Order Quantity (Q)
Annual Cost
(optimal order quantity)

33. Minimum Total Cost
The total cost curve reaches its minimum where the carrying and ordering costs are equal.

34. Total Costs Cost Adding Purchasing cost TC with PD doesn’t change EOQ
TC without PD PD
Quantity
Adding Purchasing cost
doesn’t change EOQ

35. Total Cost with Constant Carrying Costs
TCb
Total Cost
Decreasing
Price
TCc
CC a,b,c OC
EOQ
Quantity

36. Example (Example 2, p.541)
A local distributor for a national tire company expects to sell approximately 9600 steel-belted radial tires of a certain size and tread design next year. Annual carrying costs are $16 per tire, and ordering costs are $75. This distributor operate 288 days a year.
a. What is the EOQ?
b. How many times per year does the store reorder?
c. What is the length or an order cycle?
d. What is the total ordering and inventory cost?

37. Solution (Example 2, p.541)
D = 9600 tires per year, H = $16 per unit per year, S = $75
a. Q0 = {2DS/H} = {2(9600)75/16} = 300
b. Number of orders per year = D/ Q0 = 9600 / 300 = 32
c. The length of one order cycle = 1 / 32 years = 288/32 days = 9 days
d. Total ordering and inventory cost = QH/2 + DS/Q
= = 4800

38. EOQ with quantity discount
This is a variant of the EOQ model. Quantity discount is another form of economies of scale: pay less for each unit if you order more. The essential trade-off is between economies of scale and carrying cost.

39. EOQ with quantity discount
To tackle the problem, there will be a separate (TC) curve for each discount quantity price. The objective is to identify an order quantity that will represent the lowest total cost for the entire set of curves in which the solution is feasible. There are two general cases:
The holding cost is constant
The holding cost is a percentage of the purchasing price.

40. Quantity discount (constant holding cost)
Total cost per cycle
= setup cost + purchase cost + carrying cost
Setup cost = S
Purchase cost = ciQ, if the order quantity Q is in the range where unit cost is ci.
Carrying cost = (Q/2)h(Q/D)
TC = {S+ciQ+hQ2/(2D)}{D/Q} (annual cost)

41. Quantity discount (constant holding cost)
Differentiating, we obtain an optimal order quantity which is independent of the price of the good
The questions is: Is this Q0 a feasible solution to our problem?

42. Quantity discount (constant holding cost)
Solution steps:
1. Compute the EOQ.
2. If the feasible EOQ is on the lowest price curve, then it is the optimal order quantity.
3. If the feasible EOQ is on other curve, find the total cost for this EOQ and the total costs for the break points of all the lower cost curves. Compare these total costs. The point (EOQ point or break point) that yields the lowest total cost is the optimal order quantity.

43. Example
The maintenance department of a large hospital uses about 816 cases of liquid cleanser annually. Ordering costs are $12, carrying cost are $4 per case a year, and the new price schedule indicates that orders of less than 50 cases will cost $20 per case, 50 to 79 cases will cost $18 per case, 80 to 99 cases will cost $17 per case, and larger orders will cost $16 per case. Determine the optimal order quantity and the total cost.

44. Example (solutions) 1. The common EOQ: = {2(816)(12)/4} = 70
2. 70 falls in the range of 50 to 79, at $18 per case.
TC = DS/Q+ciD+hQ/2
= 816(12)/ (816) + 4(70)/2 = 14,968
3. Total cost at 80 cases per order
TC = 816(12)/ (816) + 4(80)/2 = 14,154
Total cost at 100 cases per order
TC = 816(12)/ (816) + 4(100)/2 =13,354
The minimum occurs at the break point 100. Thus order 100 cases each time

45. Quantity discount (h = rc)
Using the same argument as in the constant holding cost case, Let TCi be the total cost when the unit quantity is ci.
TCi = rciQ/2 + DS/Q + ciD
Therefore EOQ = {2DS/(rci)}
In this case, the carrying cost will decrease with the unit price. Thus the EOQ will shift to the right as the unit price decreases.

46. Quantity discount (h = rc)
Steps to identify optimal order quantity
1. Beginning with the lowest price, find the EOQs for each price range until a feasible EOQ is found.
2. If the EOQ for the lowest price is feasible, then it is the Optimal order quantity.
3. Same as step (3) in the constant carrying case.

47. Example (p.549)
Surge Electric uses 4000 toggle switches a year. Switches are priced as follows: 1 to 499 at $0.9 each; 500 to 999 at $.85 each; and 1000 or more will be at $0.82 each. It costs approximately $18 to prepare an order and receive it. Carrying cost is 18% of purchased price per unit on an annual basis. Determine the optimal order quantity and the total annual cost.

48. Solution D = 4000 per year; S = 18 ; h = 0.18 {price}
Step 1. Find the EOQ for each price, starting with the lowest price
EOQ(0.82) = {2(4000)(18)/[(0.18)(0.82)]} = 988 (Not feasible for the price range).
EOQ(0.85)= 970 (feasible for the range 500 to 999)
Step 2. Feasible solution is not on the lowest cost curve
Step 3. TC(970) = 970(.18)(.85)/ (18)/ (4000) = 3548
TC(1000) = 1000(.18)(.82)/ (18)/ (4000) = 3426
Thus the minimum total cost is 3426 and the minimum cost order size is 1000 units per order.

49. We have settled the problem of how much to order
We have settled the problem of how much to order. The next decision problem is when to order.

50. Reorder point
In the EOQ model, we assume that the delivery of goods is instantaneous. However, in real life situation, there is a time lag between the time an order is placed and the receiving of the ordered goods. We call this period of time the lead time. Demand is still occurring during the lead time and thus inventory is required to meet customer demand. This is why we need to consider when to order!

51. When to reorder We can reorder based on:
time, say weekly, monthly, etc
quantity of goods on hand

52. When to Reorder with EOQ Ordering
Reorder Point - When the quantity on hand of an item drops to this amount, the item is reordered
Safety Stock - Stock that is held in excess of expected demand due to variable demand rate and/or lead time.
Service Level - Probability that demand will not exceed supply during lead time.

53. Safety Stock Quantity Maximum probable demand during lead time
Expected demand
during lead time
ROP
Safety stock LT
Time

54. When to reorder
In general, we need to consider the following four factors when deciding when to order:
The rate of demand (usually based on a forecast)
The length of lead time.
The extent of demand and lead time variability.
The degree of stock-out risk acceptable to management.

55. When to reorder We will just cover two simple cases:
The demand rate and lead time are constant
Variable demand rate and constant lead time.

56. Reorder Point ROP Service level Risk of a stockout Probability of
no stockout
ROP
Quantity
Expected
demand
Safety
stock z
z-scale

57. ROP - constant demand and lead time
Let d be the demand rate per unit time and LT be the lead time in unit time.
The reorder point quantity is given by
ROP = expected demand during lead time + safety stock during lead time
In this case, safety stock required is 0
Thus ROP = d (LT)
Example: See Example 7 on page 551

58. ROP - variable demand rate
The lead time is constant. In this case, we need extra buffer of safety stock to insure against stock-out risk.
Since it cost money to carry safety stocks, a manager must weigh carefully the cost of carrying safety stock against the reduction in stock-out risk (provided by the safety stock)

59. ROP - variable demand rate
In other words, he needs to trade-off the cost of safety stock and the service level.
Service level = 1 - stock-out risk
ROP = Expected lead time demand + safety stock
= d (LT) + z 
Where  is the standard deviation of the lead time demand = {LT} d
Therefore ROP = d(LT) + z {LT} d

60. Example (Problem 4, p.569)
The housekeeping department of a hotel uses approximately 400 washcloths per day. The actual amount tends to vary with the number of guests on any given night. Usage can be approximated by a normal distribution that has a mean of 400 and a standard deviation of 9 washcloths per day. A linen supply company delivers towels and washcloths with a lead time of three days. If the hotel policy is to maintain a stock-out risk of 2 percent, what is the minimum number of washcloths that must be on hand at reorder time, and how much of that amount can be considered safety stock?
Solution:
d = 400, LT = 3 days, d = 9 , service level = 1 - risk = 0.98

61. Example (solution) Z = 2.055 (from normal distribution table)
Thus ROP = 400 (3) (3) d = = 1232
The safety stock is approximately 32 washcloths, providing a service level of 98%.

62. Fixed-order-interval model
In the EOQ/ROP models, fixed quantities of items are ordered at varying time interval. However, many companies ordered at fixed intervals: weekly, biweekly, monthly, etc. They order varying quantity at fixed intervals. We called this class of decision models fixed-order-interval (FOI) model.

63. FOI model Why use FOI model? Is it optimal?
What are decision variables in the FOI model?

64. FOI model
Assuming lead time is constant, we need to consider the following factors
the expected demand during the ordering interval and the lead time
the safety stock for the ordering interval and lead time
the amount of inventory on hand at the time of ordering

65. FOI model Order quantity Amount on hand Safety stock Time
Amount on hand (A) OI LT
Place order
Receive order

66. FOI model We use the following notations OI = Length of order interval
A = amount of inventory on hand
LT = lead time
d = Standard deviation of demand

67. FOI model Amount to order = expected demand during protection interval
+ Safety stock for protection interval
- amount on hand at time of placing order =

68. Example (p.560)
The following information is given for a FOI system. Determine the amount to order.
d = 30 units per day, d = 3 units per day, LT = 2 days; OI = 7 days
Amount on hand = 71 units, Desired serve level = 99%.
Solution: z = 2.33 for 99% service level.
Amount to order = 30 (7 + 2) (3) (7 + 2) = = 220