Inventory Management
Hanlin
1617_ContinuousReview.pdf
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A Continuous Review Inventory Model
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Outline
Types of Inventories Pipeline stock, cycle stock, and safety stock
A Continuous Review Inventory Model System description
Inventory concepts/equations
Replenishment policy
Performance measures
Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
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Related Readings
Operations Management (10th Edition, Prentice Hall): part of Chapter 9, pages 319-325
Operations Management (13th Edition, Pearson): part of Chapter 12, pages 508-512
Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
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Types of Inventories
Pipeline stock (due to time and distance) Things in transit from point A to point B
Cycle stock (due to economies of scale) Order in batches ==> “waves” of inventories
Safety stock (due to uncertainty) Inventories above the mean
A B
Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
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Managing Campbell’s chicken soup How does Wal-Mart manage inventory for
Campbell's chicken soup?
Consider the following inventory problem for Wal-
Mart:
Demand rate is random over time.
There is an inventory holding cost for each unit on hand
There is an ordering/setup cost for each ordering/batch production
Constant and deterministic supply lead-time
Full backlogging
There is a service level constraint (e.g., the probability of stock-out should be less than 5%)
Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
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System Description
customer demand
supplier
inventory system
Assumptions Supplier is reliable
Supply lead time is constant, L periods
Customer demand in a period is normally distributed with mean and variance
Demands in different periods are independent
When demand exceeds on-hand inventory, backlogging
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Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
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Important Inventory Concepts Net inventory = on-hand - backorders
Inventory position = net inventory + pipeline inventory
net inventory
inventory position
pipeline inventory
orders demand lead time = L
Observations:
(1) Net inventory is increased when a shipment is received and decreased when demand arrives.
(2) Inventory position is increased when an order is placed and decreased when demand arrives.
Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
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Important Inventory Concepts (Cont’d) On-order inventory / pipeline inventory / scheduled receipts
= the number of units that have been ordered but have not been received.
On-hand inventory = the number of units physically in inventory ready to serve demand.
Backorder = the total amount of demand that has not been satisfied:
All backordered demand is eventually filled, i.e., there are no lost sales.
Inventory level / Net inventory = On-hand inventory - Backorder.
Inventory position = On-order inventory + Inventory level.
Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
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An Example
In transit to store
Past orders of 4 units still need to be delivered
no backorders
In this example…
On hand = 6
Backlogs = 0 Inv. level = 6 – 0 = 6
Scheduled receipts = 3 + 4 = 7
X = Inventory Position = 6 + 7 = 13
The inventory position of the product (from the store’s perspective) is 13 units.
Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
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A Continuous Review Inventory Model
A continuous review inventory system (also called Q- system) tracks the remaining inventory continuously to determine whether it is time to reorder.
Whenever the inventory position drops to (or below) the level R, then place an order of Q units.
The control policy involves two parameters (R, Q)
Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
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(R,Q) Policy Calculations
Two parameters specify the policy: R and Q.
The value of Q can be determined using the EOQ model
If demand is certain, then
If demand is uncertain, then
R = Mean Demand During Lead-time + SAFETY STOCK
R = Demand During Lead-time
Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
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Continuous Review with Certain Demand
Time
O n
-h a n
d i n
v e n
to ry
R
TBO
L
TBO
L
TBO
L
Order received
Order received
Q
OH
Order placed
IP
Order received
Q
OH
Order placed
IP
Order received
Order placed
IP
Q
OH
Note: This is essentially an EOQ model.
Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
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Continuous Review with Uncertain Demand
Time
O n
-h a n
d i n
v e n
to ry
TBO1 TBO2 TBO3
L L L
R
Order received
Q
Order placed
Order placed
Order received
IP IP
Q
Order placed
Q
Order received
Order received
OH
Note: R is the amount of inventory we will use to satisfy the lead time demand
Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
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Use Service Level to Determine R Cycle service level (CSL, also called In-stock Probability)
measures the likelihood of not running into a stock-out by the end
of the lead time period.
CSL = Prob {demand during lead time ≤ R}
Using statistics (for the case of normal distributions): z = NORMSINV(CSL),
NORMSINV is the inverse function of the standard normal distribution function
Safety stock = zL z = The number of standard deviations needed for a given
cycle-service level. L = The standard deviation of demand during the lead time
period.
Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
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Scaling the Normal Distribution
N(=3192,=1181) N(=0,=1)
0 1250 2500 3750 5000
R -3 -1.75 -0.5 0.75 2
z
Scale down: z=(R-)/
Scale up: R = +z
Distribution function F(R) of any Normal distribution with mean and st. dev. can be found using the table of the Standard Normal distribution (z) with =0 and =1. That is, F(R) = (z) = ((R-)/). Thus we can look up z first and then convert z to R. In particular, we have
R = mean of lead time demand + z*(st. dev. of lead time demand)
Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
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A Normal Probability Distribution for an 85% In-stock Probability
Average demand
during lead time
Mean demand
during lead time
Cycle-service level = 85%
Probability of stock-out (1.0 – 0.85 = 0.15)
zL
R
Probability distribution of the lead-time demand
Let z satisfy NORMSDIST(z) = 0.85 and L be the standard deviation of lead-time demand, then safety stock = zL
Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
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Standard Normal Distribution Function Table
Examples:
NORMSDIST(2.33) = ?
NORMSINV(0.92) = ?
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How to Obtain the Lead Time Demand Distribution
Suppose the demand distribution in each period is normal with a
mean and a standard deviation .
The lead time is L periods and the demand distributions are
independent and identical across period.
Then the distribution for the demand during the lead time has a
mean L = L and a standard deviation LL
Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
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An Example
= 15
+ = 75
Demand for week 1
L = 26
L = 225 Demand for 3-week lead time
+ 75
Demand for week 2
= 15
= 75
Demand for week 3
= 15
Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
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Calculating Total Costs in a Continuous Review Model
Total policy-related costs for the continuous review system is the
sum of three cost components:
= Annual cycle inventory holding cost + annual ordering cost
+ annual safety stock holding cost
= LHzK
Q
D H
Q )()(
2
where D: Mean annual demand Q: Economic order quantity K: Fixed ordering cost H: Annual inventory carrying cost
Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
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Example 12.4: Finding R and Total Costs 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. (Recall EOQ = 75, K = $45 per order, and H = $15 per unit per year from Example 12.2) What is the optimal R (rounded to the nearest integer)?
Answer:
L = L = 5 2 = 7.1
Safety stock = zL = 1.29(7.1) = 9.16 or 9 units Reorder point = L + safety stock = 2(18) + 9 = 45 units
Demand distribution for lead time must be developed:
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Example 12.4: Finding R and Total Costs What is the total policy-related cost of the Q system (keep two digits after the decimal point)?
Answer:
Safety stock = zL = 1.29(7.1) = 9.16 or 9 units Reorder point = L + safety stock = 2(18) + 9 = 45 units
C = ($15) + ($45) + 9($15) 75
2
936
75
C = $562.50 + $561.60 + $135 = $1259.10
(Recall Q = EOQ = 75)
Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
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Calculating Performance Measures for a (R,Q) Policy
Suppose we use a (R,Q) inventory policy, with given parameters R and Q.
What is the CSL this policy achieves?
L
LRNORMSDISTCSL
Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
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Example: Finding (Q, R) and the CSL A regional warehouse purchases hand tools from various suppliers and then distributes to retailers in the region. The warehouse operates 5 days a week, 52 weeks per year. The following data are estimated for 3/8-inch hand drills with double insulation and variable speeds:
Average daily demand = 100 drills Standard deviation of daily demand = 30 drills Lead time L = 3 days
Holding cost H = $9.40 per unit per year
Ordering cost K = $35 per order
Cycle-service level = 92%
(a) What order quantity Q (rounded to the nearest integer) should be used?
(b) What reorder point R (rounded to the nearest integer) should be used?
(c) If on-hand inventory is 40 units, one order for 440 drills is pending, and no backorder exists, should a new order be placed?
(d) Suppose the manager has chosen R = 400. What will CSL be? Keep three digits after the decimal point.
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Example: Solution a) Annual demand is D = 5*52 *100 = 26,000 drills a year
The order quantity is
b) The standard deviation for lead-time demand is
At 92% CSL, we have z = 1.41 (from Normal distribution Table). Therefore, safety stock = zL = 1.41*52 73 drills and reorder point R = 100(3) + 73 = 373 drills.
c) IP = OH +SR - BO = 40 + 440 -0 = 480 drills > R, so do not place a new order
d)
drills H
DK EOQ 440167,193
40.9$
)35)($000,26(22
drillsLL 52330
L
LRNORMSDISTCSL
.973.0)923.1( 52
300400
NORMSDISTNORMSDIST
Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
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Summary
Inventory Concepts and Equations
Continuous Review Inventory Models Policy calculation for a given service level requirement
Service level evaluation for a given control policy
Classes 16&17-Continuous Review Inventory Model MGT 303 Prof. Yang
