1. Continuous & periodic review systems safety stocks

2. EOQ for constant Demand & LeadTime
Order
placed IP Q OH
Order
received Q OH
placed IP
Time
On-hand inventory
Order
received R
TBO L
TBO L

3. Impact of lead time and uncertainty in demand
Lead time has NO impact if the demand is deterministic and at a constant rate.
Uncertainty in the demand creates the need for safety stock
Lead time under uncertain demand requires even a larger safety stock!

4. EOQ for Uncertain Demand and Constant Lead Time
Order
received Q
placed IP OH
Time
On-hand inventory R
TBO1
TBO2
TBO3 L

5. Choosing an Appropriate Service-Level Policy
Service level (Cycle-service level): The desired probability of not running out of stock in any one ordering cycle, which begins at the time an order is placed and ends when it arrives.
Protection interval: The period over which safety stock must protect the user from running out (in this case, it will be the leadtime period).
Reorder point (R) = DL + Safety stock (SS)
Safety stock (SS) = zsL
z = The number of standard deviations needed for a given cycle-service level.
sL=Standard deviation of the demand during lead time
=The average demand during the lead time period DL

6. Finding Safety Stock With a normal Probability Distribution for an 85% Cycle-Service Level
Average demand during lead time
Average demand (D) during lead time
Cycle-service level = 85%
Probability of stockout
(1.0 – 0.85 = 0.15)
zL R

7. Finding Safety Stock and R
Records show that the demand for dishwasher detergent during the lead time is normally distributed, with an average of 250 boxes and L = 22. What safety stock should be carried for a 99 percent cycle-service level? What is R?
Safety stock (SS) = zsL
= 2.33(22) = 51.3
= 51 boxes
Reorder point = DL + SS =
= 301 boxes
2.33 is the number of standard deviations, z, to the right of average demand during the lead time that places 99% of the area under the curve to the left of that point.

8. In Class Example
Suppose that the demand for an item during the lead time period is normally distributed with and an average of 85 and a standard deviation of 40.
Find the safety stock and reorder point for a service level of 95%
How much reduction is safety stock will result if the desired service level is reduced to 85%

9. Development of Demand Distributions for the Lead Time+ 75
Demand for week 1 + 75
Demand for week 2
st = 15 = 75
Demand for week 3
st = 15
st = 26
225
Demand for 3-week lead time

10. Continuous Review Systems
Selecting the reorder point with variable demand and constant lead time
Reorder point = Average demand during lead time + Safety stock
= dL + safety stock
Where
d = average demand per week (or day or months)
L = constant lead time in weeks (or days or months)

11. Demand During Lead Time
Specify mean and standard deviation
Standard deviation of demand during lead time
σdLT = σd2L = σd L
Safety stock and reorder point
Safety stock = zσdLT
where
z = number of standard deviations needed to achieve the cycle-service level
σdLT = stand deviation of demand during lead time
Reorder point R = dL + safety stock

12. Continuous Review Systems General Cost Equation
Calculating total systems costs
Total cost = Annual cycle inventory holding cost + Annual ordering cost + Annual safety stock holding cost
C = (H) (S) + (H) (Safety stock) Q 2 D
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13. Finding Safety Stock and R
Suppose that the average demand for bird feeders is 18 units per week with a standard deviation of 5 units. The lead time is constant at 2 weeks. Determine the safety stock and reorder point for a 90 percent cycle-service level. What is the total cost of the Q system? (t = 1 week; d = 18 units per week; L = 2 weeks)
Demand distribution for lead time must be developed:
sL = st L = = 7.1
Safety stock = zsL = 1.28(7.1) = 9.1 or 9 units
Reorder point = dL + safety stock = 2(18) + 9 = 45 units
C = ($15) ($45) + 9($15) 75 2
936
C = $ $ $135 = $

14. Class Example:
The following info is available for the purchase of kitty litter:
Demand: 100 bags/week with a standard deviation of 10 bags/week (assume 50 weeks/year)
Price: $10/bag
Ordering costs: $100/order
Annual Holding Costs: 10% of price
Desired service level: 99%
Lead time: 4 weeks
What is the Order Quantity and the Reorder Point that assures this service level while minimizing inventory costs. What is the minimum inventory costs?

15. Reorder Point for Variable Demand and Lead Time
Often the case that both are variable
The equations are more complicated
Safety stock = zσdLT
R = (Average weekly demand  Average lead time) + Safety stock
= dL + Safety stock
where
σdLT = Lσd2 + d2σLT2

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Solved Problem
Grey Wolf Lodge is a popular 500-room hotel in the North Woods. Managers need to keep close tabs on all room service items, including a special pine-scented bar soap. The daily demand for the soap is 275 bars, with a standard deviation of 30 bars. Ordering cost is $10 and the inventory holding cost is $0.30/bar/year. The lead time from the supplier is 5 days, with a standard deviation of 1 day. The lodge is open 365 days a year.
a. What is the economic order quantity for the bar of soap?
b. What should the reorder point be for the bar of soap if management wants to have a 99 percent cycle-service level?
c. What is the total annual cost for the bar of soap, assuming a Q system will be used?
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17. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solved Problem
SOLUTION
a. We have D = (275)(365) = 100,375 bars of soap; S = $10; and H = $0.30. The EOQ for the bar of soap is
EOQ = =
2DS H
2(100,375)($10)
$0.30
= 6,691, = 2, or 2,587 bars
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18. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solved Problem
b. We have d = 275 bars/day, σd = 30 bars, L = 5 days, and σLT = 1 day.
σdLT = Lσd2 + d2σLT2 =
(5)(30)2 + (275)2(1)2 = bars
Consult the body of the Normal Distribution appendix for The closest value is , which corresponds to a z value of We calculate the safety stock and reorder point as follows:
Safety stock = zσdLT =
(2.33)(283.06) = or 660 bars
Reorder point = dL + Safety stock =
(275)(5) = 2,035 bars
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19. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solved Problem
c. The total annual cost for the Q system is
C = (H) (S) + (H)(Safety stock) Q 2 D
C = ($0.30) ($10) + ($0.30)(660) = $974.05
2,587 2
100,375
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20. Periodic Review System (P)
Fixed interval reorder system or periodic reorder system
Four of the original EOQ assumptions maintained
No constraints are placed on lot size
Holding and ordering costs
Independent demand
Lead times are certain
Order is placed to bring the inventory position up to the target inventory level, T, when the predetermined time, P, has elapsed
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21. Periodic Review System (P)
Time
On-hand inventory T
Order
placed
received IP OH Q1 Q2 Q3
IP3
IP1
IP2 L P
Protection interval
Figure – P System When Demand Is Uncertain
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22. How Much to Order in a P System
EXAMPLE
A distribution center has a backorder (BO) for five 36-inch color TV sets. No inventory is currently on hand (OH), and now is the time to review. How many should be reordered if T = 400 and no receipts are scheduled (SR)?
SOLUTION
IP = OH + SR – BO
= – 5 = –5 sets
T – IP =
400 – (–5) = 405 sets
That is, 405 sets must be ordered to bring the inventory position up to T sets.
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23. Periodic Review System
Selecting the period of time between reviews (P)
The order-up-to level (T) when demand is variable and lead time is constant will be equal to the average demand during the protection period (P+L) + Safety Stock
T = d(P + L) + safety stock for protection interval
Safety stock = zσP + L , where σP + L =
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

24. Finding Safety Stock and R Continuous Review Model Example
Suppose that the average demand for bird feeders is 18 units per week with a standard deviation of 5 units. The lead time is constant at 2 weeks. Determine the safety stock and reorder point for a 90 percent cycle-service level. What is the total cost of the Q system? (t = 1 week; d = 18 units per week; L = 2 weeks)
Demand distribution for lead time must be developed:
sL = st L = = 7.1
Safety stock = zsL = 1.28(7.1) = 9.1 or 9 units
Reorder point = dL + safety stock = 2(18) + 9 = 45 units
C = ($15) ($45) + 9($15) 75 2
936
C = $ $ $135 = $

25. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Calculating P and T
What is the equivalent P system to the bird feeder example? Recall that demand for the bird feeder is normally distributed with a mean of 18 units per week and a standard deviation in weekly demand of 5 units. The lead time is 2 weeks, and the business operates 52 weeks per year. The Q system calls for an EOQ of 75 units and a safety stock of 9 units for a cycle-service level of 90 percent.
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26. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Calculating P and T
SOLUTION
We first define D and then P. Here, P is the time between reviews, expressed in weeks because the data are expressed as demand per week:
D =
(18 units/week)(52 weeks/year) = 936 units
P = (52) =
EOQ D
(52) = 4.2 or 4 weeks 75
936
With d = 18 units per week, an alternative approach is to calculate P by dividing the EOQ by d to get 75/18 = 4.2 or 4 weeks. Either way, we would review the bird feeder inventory every 4 weeks.
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27. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Calculating P and T
We now find the standard deviation of demand over the protection interval (P + L) = 6:
Before calculating T, we also need a z value. For a 90 percent cycle-service level z = The safety stock becomes
Safety stock = zσP + L =
1.28(12.25) = or 16 units
We now solve for T:
T = Average demand during the protection interval + Safety stock
= d(P + L) + safety stock
= (18 units/week)(6 weeks) + 16 units = 124 units
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28. Periodic Review System
Use simulation when both demand and lead time are variable
Total costs for the P system are the sum of the same three cost elements as in the Q system
Order quantity and safety stock are calculated differently
C = (18 units/week)*(4 weeks)/2*(15) + 936/(18*4)*(45) + (15)*1.28*(12.25)
C = 36* * *16 = = $1,365
C = (H) (S) + HzσP + L dP 2 D
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29. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
In Class Example
Discount Appliance Store has the following information:
Demand = 10 units/wk (assume 52 weeks per year) = 520
EOQ = 62 units (with reorder point system)
Lead time (L) = 3 weeks
Standard deviation in weekly demand = 8 units
Cycle-service level of 70% (z = )
Choose the Reorder interval P such as this system is approximates the EOQ model.
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30. Multi-period Inventory Models: Fixed Time Period (Fixed-Order- Interval) Models

31. Fixed-Order-Interval Model
Orders are placed at fixed time intervals
Order quantity for next interval? (inventory is brought up to target amount, amount ordered varies)
Suppliers might encourage fixed intervals
Requires only periodic checks of inventory levels (no continous monitoring is required)
Risk of stockout between intervals

32. Inventory Level in a Fixed Period System
Various amounts (Qi) are ordered at regular time intervals (p) based on the quantity necessary to bring inventory up to target maximum p Q1 Q2 Q3 Q4
Target maximum
Time
d Inventory

33. Fixed-Interval Benefits
Tight control of inventory items
Items from same supplier may yield savings in:
Ordering
Packing
Shipping costs
May be practical when inventories cannot be closely monitored

34. Fixed-Interval Disadvantages
Requires a larger safety stock
Increases carrying cost
Costs of periodic reviews

35. Fixed-Time Period Model with Safety Stock Formula
q = Average demand + Safety stock – Inventory currently on hand

36. Fixed-Time Period Model: Determining the Value of sT+L
The standard deviation of a sequence of random events equals the square root of the sum of the variances 20

37. Order Quantity for a Periodic Inventory System
Q = d(tb + L) + zd T + L - I
where
d = average demand rate
T = the fixed time between orders
L = lead time
 d = standard deviation of demand
zd tb + L = safety stock
I = inventory level
z = the number of standard deviations
for a specified service level

38. Fixed-Period Model with Variable Demand (Example 1)
d = 6 bottles per day
sd = 1.2 bottles
tb = 60 days
L = 5 days
I = 8 bottles
z = 1.65 (for a 95% service level)
Q = d(tb + L) + zd tb + L - I
= (6)(60 + 5) + (1.65)(1.2)
= bottles

39. Fixed-Time Period Model with Variable Demand (Example 2)(1 of 3)
Given the information below, how many units should be ordered?
Average daily demand for a product is 20 units. The review period is 30 days, and lead time is 10 days. Management has set a policy of satisfying 96 percent of demand from items in stock. At the beginning of the review period there are 200 units in inventory. The standard deviation of daily demand is 4 units. 21

40. Fixed-Time Period Model with Variable Demand (Example 2)(2 of 3)
So, by looking at the value from the Table, we have a probability of , which is given by a z = 1.75 22

41. Fixed-Time Period Model with Variable Demand (Example 2) (3 of 3)
So, to satisfy 96 percent of the demand, you should place an order of 645 units at this review period 23

42. Miscellaneous Systems: Optional Replenishment System
Maximum Inventory Level, M
Actual Inventory Level, I
q = M - I I M
Q = minimum acceptable order quantity
If q > Q, order q, otherwise do not order any. 24

43. Comparative Advantages
Primary advantages of P systems
Convenient
Orders can be combined
Only need to know IP when review is made
Primary advantages of Q systems
Review frequency may be individualized
Fixed lot sizes can result in quantity discounts
Lower safety stocks
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48. Single Period Model
Assume that you want to have a certain level of confidence that you won’t run out of stock and that the demand follows a normal distribution, then the inventory level you should carry will be equal to:
Q = D + zs

49. Example
So if the demand for newspapers on Monday’s is normally distributed with a mean of 90 and standard deviation of 10, and the newsboy wants to be 80% certain that he/she will not run out of papers, then the number of papers he/she should order will be equal to:
Q = D + zs
Q = * 10 = 98.4 = 99 papers

50. And to make it even more interesting
If we have the following cost data:
Cost per unit of overestimating demand
Cost per unit of underestimating demand
Then:
Probability of stockouts <= Cu / (Cu + Co)

51. Example continued
If we assume that the newspaper boy pays 20 cents per paper and he sells it for 50 cents. How many newspapers should he order if the demand is normally distributed with a mean of 90 and standard deviation of 10?
Cost of underestimating (Lost sales)= = .3
Cost of overestimation (stock piling) = .2
Probability of stock outs <= .3/(.2+.3) <= .6 <= 60%
Z = .253
Q = * 10 = = 93 newspapers

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In Class Example
Assume you are helping a Christmas tree retailer determine how many trees to order for this year’s season. Assuming that you know from past experience that the average demand for Christmas trees in his area is 500 but that the demand over the past 25 years has varied depending on the economy and the offers on plastic trees. The standard deviation of the demand is 100 trees. If this person can buy each tree at an average cost of $5 and sell them at $50, then how many trees would you recommend he orders?
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