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Transcript of 12 – 1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Inventory Management...
12 – 1Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Inventory Management12
For Operations Management, 9e by Krajewski/Ritzman/Malhotra © 2010 Pearson Education
12 – 2
Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e
© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 0745812-2
Learning Objectives Know the types of inventory and their associated costs Be able to do an ABC analysis for a list of inventory items Be able to determine the Economic Order Quantity (EOQ) for an
inventory item and adapt this concept to volume pricing options Be able to determine the Economic Production Lot Size (ELS)
for an inventory item based on its demand and production rates. (discussed in Supplement D in your text).
Be able to determine reorder points and appropriate safety stock levels for a desired level of service
Know the differences between ELS and EOQ lot-sizing rules Know the differences between continuous review and periodic
review of inventory strategies and where each is best used. Be able to list the various approaches for reducing required
inventory levels
12 – 3
3
Functions of Inventory
Meet customer demands, provide selectionDe-couple production from distribution Decrease costs (all types)De-couple production processes Manage input demand (services, backlogs)Uncertainty protection
12 – 4
4
Disadvantages of Inventory(All measured by increased cost)
Increased costs for storage, handling, tracking, and protection
Hides production problemsRisk of deterioration or obsolescenceRisk of lost business (demand inventory)
12 – 5
5
Types of Inventory – Where Did It Come From?Cycle inventory –> result of lot size = (Q/2)+SSRaw –> fixed order sizesWIP and FG –> fixed production batch sizesSafety stock –> just in case something goes wrong –
all typesDemand/Backorders -> insufficient capacityAnticipation inventory –> all typesPipeline –> demand rate times transit time = d×LVendor Managed Inventory (VMI)Maintenance/Repair/Operating Supplies
12 – 6
6
Inventory Notation and Definitions IP = inventory position = OH + SR – BO T = target inventory position OH = on-hand inventory SR = scheduled receipts (open orders) BO = back orders (delayed delivery) ROP = reorder point (inventory level when order is placed) Q = order quantity TBO = time between orders for continuous review P = fixed time between orders for periodic review L = lead time from order until delivery Protection interval = P + L ADDLT = average demand during lead time SO = stock out (no material to satisfy demand) SS = safety stock (cushion to prevent stock out)
12 – 7
12-7
Inventory
Process stage
Demand Type
Number and Value Other
Raw Material WIP
Finished Goods
Independent Dependent
A Items B Items C Items
Maintenance Operating
Physical Inventory Classifications
12 – 8
12-8
Demand Inventory ClassificationsWaiting LinesBooked AppointmentsReservationsBacklogs
12 – 9
12-9
Pressures on inventory
Pressure for lower inventory• Inventory investment
• Inventory holding cost
Pressure for higher inventory• Customer service
• Other costs related to inventory
12 – 10
12-10
Enables effective focus, prioritizationDivides on-hand inventory into 3 classes
A class, B class, C class
Basis is usually annual dollar volume (investment) $ volume = Annual demand x Unit cost
Policies based on ABC analysis Develop class A suppliers more Establish tighter physical control of A items Forecast A items more carefully
ABC Analysis
12 – 11
12-11
10 20 30 40 50 60 70 80 90 100
Percentage of items
Pe
rce
nta
ge
of
do
llar
va
lue
100 —
90 —
80 —
70 —
60 —
50 —
40 —
30 —
20 —
10 —
0 —
Class C
Class A
Class B
Class % $Vol % Items
A 80 20B 15 30C 5 50
ABC Analysis
12 – 12
12-12
Inventory Costs
Holding costs - associated with holding or “carrying” inventory over time
Ordering costs - associated with costs of placing order and receiving goods
Setup costs - cost to prepare a machine or process for manufacturing an order
Transportation costs – associated with distribution (pipeline inventory).
12 – 13
12-13
Holding (Carrying) Costs
Obsolescence InsuranceExtra staffing for handling, management InterestPilferageDamageWarehousingTaxesTracking
IBM Tracking RFIDEtc.
12 – 14
12-14
Inventory Holding Costs(Approximate Ranges)
Category
Housing costs (building rent, depreciation, operating cost, taxes, insurance)
Material handling costs (equipment, lease or depreciation, power, operating cost)
Labor cost from extra handling
Investment costs (borrowing costs, taxes, and insurance on inventory)
Pilferage, scrap, and obsolescence
Overall carrying cost
Cost as a % of Inventory Value
6%(3 - 10%)
3%(1 - 3.5%)
3%(3 - 5%)
11%(6 - 24%)
3% (2 - 5%)
26%
12 – 15
12-15
Ordering Costs
SuppliesFormsOrder placement (submitting,
tracking, receiving)Clerical supportPackaging waste disposalEtc.
12 – 16
12-16
Setup Costs
Clean-up costsRe-tooling costsAdjustment costsInitial yield lossesDocumentation and
trackingWIP managementRoutine maintenanceEtc.
12 – 17
12-17
Inventory Management
¨ Expected demand (accurate forecasting)¨ Amounts on hand¨ Amounts on order¨ Appropriate timing and size of the reorder
quantities¨ Acquisition and holding costs
Effective inventory management requires information about:
12 – 18
12-18
Continuous review models (Q) Economic order quantity (EOQ) Economic production order
size (ELS) Quantity discount
Periodic review models (P) Probabilistic models
Help answer the inventory planning questions!
Help answer the inventory planning questions!
© 1984-1994 T/Maker Co.
Inventory Models
12 – 19Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Economic Order Quantity
The lot size, Q, that minimizes total annual inventory holding and ordering costs
Five assumptions1. Demand rate is constant and known with certainty
2. No constraints are placed on the size of each lot
3. The only two relevant costs are the inventory holding cost and the fixed cost per lot for ordering or setup
4. Decisions for one item can be made independently of decisions for other items
5. The lead time is constant and known with certainty
12 – 20Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Economic Order Quantity
Don’t use the EOQ Make-to-order strategy Order size is constrained
Modify the EOQ Quantity discounts Replenishment not instantaneous
Use the EOQ Make-to-stock Carrying and setup costs are known and
relatively stable
12 – 21Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Calculating EOQ
Inventory depletion (demand rate)
Receive order
1 cycle
On
-han
d i
nve
nto
ry (
un
its)
Time
Q
Averagecycleinventory
Q
2
Figure 12.2 – Cycle-Inventory Levels
12 – 22Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Calculating EOQ
Annual holding cost
Annual holding cost = (Average cycle inventory)
(Unit holding cost)
Annual ordering cost
Annual ordering cost = (Number of orders/Year) (Ordering or setup costs)
Total annual cycle-inventory cost
Total costs = Annual holding cost + Annual ordering or setup cost
12 – 23Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
An
nu
al c
ost
(d
olla
rs)
Lot Size (Q)
Holding cost
Ordering cost
Total cost
Calculating EOQ
Figure 12.3 – Graphs of Annual Holding, Ordering, and Total Costs
12 – 24Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Calculating EOQ
Total annual cycle-inventory cost
whereC = total annual cycle-inventory costQ = lot sizeH = holding cost per unit per yearD = annual demandS = ordering or setup costs per lot
C = (H) + (S)Q
2DQ
12 – 25Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
The Cost of a Lot-Sizing Policy
EXAMPLE 12.1 A museum of natural history opened a gift shop which
operates 52 weeks per year. Managing inventories has become a problem. Top-selling SKU is a bird feeder. Sales are 18 units per week, the supplier charges $60 per
unit. Ordering cost is $45. Annual holding cost is 25 percent of a feeder’s value. Management chose a 390-unit lot size. What is the annual cycle-inventory cost of the current policy
of using a 390-unit lot size? Would a lot size of 468 be better?
12 – 26Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
The Cost of a Lot-Sizing Policy
SOLUTION
We begin by computing the annual demand and holding cost as
D =
H =
C = (H) + (S)Q
2DQ
The total annual cycle-inventory cost for the alternative lot size is
= ($15) + ($45)
= $2,925 + $108 = $3,033
3902
936390
The total annual cycle-inventory cost for the current policy is
(18 units/week)(52 weeks/year) = 936 units
0.25($60/unit) = $15
C = ($15) + ($45) = $3,510 + $90 = $3,600468
2936468
12 – 27Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
The Cost of a Lot-Sizing Policy
3000 –
2000 –
1000 –
0 –
| | | | | | | |
50 100 150 200 250 300 350 400
Lot Size (Q)
An
nu
al c
ost
(d
olla
rs)
CurrentQ
Currentcost
Lowestcost
Best Q (EOQ)
Total cost = (H) + (S)
Q
2DQ
Figure 12.4 – Total Annual Cycle-Inventory Cost Function for the Bird Feeder
Ordering cost = (S)DQ
Holding cost = (H)Q
2
12 – 28Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Calculating EOQ
The EOQ formula:
EOQ = 2DSH
Time between orders
TBOEOQ = (12 months/year)EOQ
D
12 – 29Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Finding the EOQ, Total Cost, TBO
EXAMPLE 12.2
For the bird feeders in Example 12.1, calculate the EOQ and its total annual cycle-inventory cost. How frequently will orders be placed if the EOQ is used?
SOLUTION
Using the formulas for EOQ and annual cost, we get
EOQ = =2DS
H= 74.94 or 75 units2(936)(45)
15
12 – 30Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Finding the EOQ, Total Cost, TBO
Figure 12.5 shows that the total annual cost is much less than the $3,033 cost of the current policy of placing 390-unit orders.
Figure 12.5 – Total Annual Cycle-Inventory Costs Based on EOQ Using Tutor 12.2
12 – 31Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Finding the EOQ, Total Cost, TBO
When the EOQ is used, the TBO can be expressed in various ways for the same time period.
TBOEOQ =EOQ
D
TBOEOQ = (12 months/year)EOQ
D
TBOEOQ = (52 weeks/year)EOQ
D
TBOEOQ = (365 days/year)EOQ
D
= = 0.080 year75
936
= (12) = 0.96 month75
936
= (52) = 4.17 weeks75
936
= (365) = 29.25 days75
936
12 – 32Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Noninstantaneous Replenishment
Maximum cycle inventory Item used or sold as it is completedUsually production rate, p, exceeds the demand
rate, d, so there is a buildup of (p – d) units per time period
Both p and d expressed in same time intervalBuildup continues for Q/p days
12 – 33Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Production quantityQ
Maximum inventoryImax
Production and demand
Demand only
TBO
p – d
Demand during production interval
On-
hand
inve
ntor
y
Time
Noninstantaneous Replenishment
Figure D.1 – Lot Sizing with Noninstantaneous Replenishment
12 – 34Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Noninstantaneous Replenishment
Cycle inventory is no longer Q/2, it is Imax /2
Maximum cycle inventory is:
where
p = production rate
d = demand rate
Q = lot size
pdp
QdppQ
Imax
12 – 35Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Noninstantaneous Replenishment
D is annual demand and Q is lot sized is daily demand; p is daily production rate
Total annual cost = Annual holding cost + Annual ordering or setup cost
SQD
Hp
dpQS
QD
HI
C
22max
12 – 36Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Noninstantaneous Replenishment
Economic Production Lot Size (ELS): optimal lot size Derived by calculus Because the second term is greater than 1, the
ELS results in a larger lot size than the EOQ
dpp
HDS
ELS
2
12 – 37Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Finding the Economic Production Lot Size
EXAMPLE D.1
A plant manager of a chemical plant must determine the lot size for a particular chemical that has a steady demand of 30 barrels per day. The production rate is 190 barrels per day, annual demand is 10,500 barrels, setup cost is $200, annual holding cost is $0.21 per barrel, and the plant operates 350 days per year.
a. Determine the economic production lot size (ELS)b. Determine the total annual setup and inventory holding cost for this itemc. Determine the time between orders (TBO), or cycle length, for the ELSd. Determine the production time per lot
What are the advantages of reducing the setup time by 10 percent?
12 – 38Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Finding the Economic Production Lot Size
SOLUTION
a. Solving first for the ELS, we get
dpp
HDS
2ELS
barrels 4,873.4
30190190
210200500102
.$
$,
b. The total annual cost with the ELS is
SQD
Hp
dpQC
2
20048734
50010210
19030190
248734
$.,
,.$
.,
828619143091430 .$.$.$
12 – 39Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Finding the Economic Production Lot Size
c. Applying the TBO formula to the ELS, we get
days/year 350ELS
TBOELS D
days 162 or 162.4
d. The production time during each cycle is the lot size divided by the production rate:
p
ELS
35050010
48734,
.,
days 26 or 25.6190
48734
.,
12 – 40Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Managerial Insights
TABLE 12.1 | SENSITIVITY ANALYSIS OF THE EOQ
Parameter EOQ Parameter Change
EOQ Change
Comments
Demand ↑ ↑ Increase in lot size is in proportion to the square root of D.
Order/Setup Costs ↓ ↓
Weeks of supply decreases and inventory turnover increases because the lot size decreases.
Holding Costs ↓ ↑ Larger lots are justified when holding
costs decrease.
2DSH
2DSH
2DSH
12 – 41Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Inventory Control Systems
Continuous review (Q) system Reorder point system (ROP) and fixed order
quantity (FOQ) system — FOQ often = EOQ For independent demand items Tracks inventory position (IP) Includes scheduled receipts (SR), on-hand
inventory (OH), and back orders (BO)
Inventory position = On-hand inventory + Scheduled receipts – Backorders
IP = OH + SR – BO
12 – 42Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Selecting the Reorder Point
Time
On
-han
d in
ven
tory
TBO TBO
L L
TBO
L
Orderplaced
Orderplaced
Orderplaced
IP IPIP
R
Q QQ
OH OHOH
Orderreceived
Orderreceived
Orderreceived
Orderreceived
Figure 12.6 – Q System When Demand and Lead Time Are Constant and Certain
12 – 43Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Placing a New Order
EXAMPLE 12.3
Demand for chicken soup at a supermarket is always 25 cases a day and the lead time is always 4 days. The shelves were just restocked with chicken soup, leaving an on-hand inventory of only 10 cases. No backorders currently exist, but there is one open order in the pipeline for 200 cases. What is the inventory position? Should a new order be placed?
SOLUTION
R = Total demand during lead time = (25)(4) = 100 cases
= 10 + 200 – 0 = 210 cases
IP = OH + SR – BO
12 – 44Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
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
whered = average demand per week (or day or months)L = constant lead time in weeks (or days or months)
12 – 45Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Continuous Review Systems
Time
On
-han
d in
ven
tory
TBO1 TBO2 TBO3
L1 L2 L3
R
Orderreceived
Q
Orderplaced
Orderplaced
Orderreceived
IP IP
Q
Orderplaced
Q
Orderreceived
Orderreceived
0
IP
Figure 12.7 – Q System When Demand Is Uncertain
12 – 46Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Reorder Point
1. Choose an appropriate service-level policy Select service level or cycle service level Protection interval
2. Determine the demand during lead time probability distribution
3. Determine the safety stock and reorder point levels
12 – 47Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Demand During Lead Time
Specify mean and standard deviationStandard deviation of demand during lead time
σdLT = σd2L = σd L
Safety stock and reorder point
Safety stock = zσdLT
wherez = number of standard deviations needed to achieve the cycle-service levelσdLT = standard deviation of demand during lead time
Reorder point = R = dL + safety stock
12 – 48Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
σt = 15
+75
Demand for week 1
σt = 25.98
225Demand for 3-week lead time
+75
Demand for week 2
σt = 15
=75
Demand for week 3
σt = 15
Demand During Lead Time
Figure 12.8 – Development of Demand Distribution for the Lead Time
12 – 49Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Demand During Lead Time
Average demand
during lead time
Cycle-service level = 85%
Probability of stockout(1.0 – 0.85 = 0.15)
zσdLT
R
Figure 12.9 – Finding Safety Stock with a Normal Probability Distribution for an 85 Percent Cycle-Service Level
12 – 50Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Reorder Point for Variable Demand
EXAMPLE 12.4
Let us return to the bird feeder in Example 12.2. The EOQ is 75 units. Suppose that the average demand is 18 units per week with a standard deviation of 5 units. The lead time is constant at two weeks. Determine the safety stock and reorder point if management wants a 90 percent cycle-service level.
12 – 51Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Reorder Point for Variable Demand
SOLUTION
In this case, σd = 5, d = 18 units, and L = 2 weeks, so σdLT = σd L = 5 2 = 7.07. Consult the body of the table in the Normal Distribution appendix for 0.9000, which corresponds to a 90 percent cycle-service level. The closest number is 0.8997, which corresponds to 1.2 in the row heading and 0.08 in the column heading. Adding these values gives a z value of 1.28. With this information, we calculate the safety stock and reorder point as follows:
Safety stock = zσdLT = 1.28(7.07) = 9.05 or 9 units
2(18) + 9 = 45 unitsReorder point = dL + Safety stock =
12 – 52Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Reorder Point for Variable Demand and Lead Time
Often the case that both are variableThe equations are more complicated
Safety stock = zσdLT
whered = Average weekly (or daily or monthly) demandL = Average lead timeσd = Standard deviation of weekly (or daily or monthly) demandσLT = Standard deviation of the lead timeσdLT = Lσd
2 + d2σLT2
R = (Average weekly demand Average lead time) + Safety stock
= dL + Safety stock
12 – 53Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Reorder Point
EXAMPLE 12.5
The Office Supply Shop estimates that the average demand for a popular ball-point pen is 12,000 pens per week with a standard deviation of 3,000 pens. The current inventory policy calls for replenishment orders of 156,000 pens. The average lead time from the distributor is 5 weeks, with a standard deviation of 2 weeks. If management wants a 95 percent cycle-service level, what should the reorder point be?
12 – 54Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Reorder Point
SOLUTION
From the Normal Distribution appendix for 0.9500, the appropriate z value = 1.65. We calculate the safety stock and reorder point as follows:
σdLT = Lσd2 + d2σLT
2 =
We have d = 12,000 pens, σd = 3,000 pens, L = 5 weeks, and σLT = 2 weeks
(5)(3,000)2 + (12,000)2(2)2
= 24,919.87 pens
Safety stock = zσdLT =
Reorder point = dL + Safety stock =
(1.65)(24,919.87) = 41,117.79 or 41,118 pens
(12,000)(5) + 41.118 = 101,118 pens
12 – 55Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Continuous Review (Q) Systems
Two-Bin system Visual system An empty first bin signals the need to place an order
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
2DQ
12 – 56Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
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
12 – 57Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Periodic Review System (P)
P P
T
L L L
Protection interval
Time
On
-han
d in
ven
tory
IP3
IP1
IP2
OrderplacedOrderplaced
Orderplaced
Orderreceived
Orderreceived
Orderreceived
IP IPIP
OH OHQ1
Q2
Q3
Figure 12.10 – P System When Demand Is Uncertain
12 – 58Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
How Much to Order in a P System
EXAMPLE 12.6
A distribution center has a backorder for five 36-inch color TV sets. No inventory is currently on hand, and now is the time to review. How many should be reordered if T = 400 and no receipts are scheduled?
SOLUTION
IP = OH + SR – BO
That is, 405 sets must be ordered to bring the inventory position up to T sets.
= 0 + 0 – 5 = –5 sets
T – IP = 400 – (–5) = 405 sets
12 – 59Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Periodic Review System
Selecting the time between reviews, choosing P and T
T = d(P + L) + safety stock for protection interval
Selecting T when demand is variable and lead time is constant
IP covers demand over a protection interval of P + L
The average demand during the protection interval is d(P + L), or
Safety stock = zσP + L , where σP + L = LPd
12 – 60Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Calculating P and T
EXAMPLE 12.7
Again, let us return 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 developed in Example 12.4 called for an EOQ of 75 units and a safety stock of 9 units for a cycle-service level of 90 percent. What is the equivalent P system? Answers are to be rounded to the nearest integer.
12 – 61Copyright © 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
75936
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.
12 – 62Copyright © 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 = 1.28. The safety stock becomes
Safety stock = zσP + L = 1.28(12.25) = 15.68 or 16 units
We now solve for T:
= (18 units/week)(6 weeks) + 16 units = 124 units
T = Average demand during the protection interval + Safety stock
= d(P + L) + safety stock
units 12.2565 LPdLP
12 – 63Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Periodic Review System
Use simulation when both demand and lead time are variable
Suitable to single-bin systems 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 = (H) + (S) + HzσP + L
dP2
DdP
12 – 64Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
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
12 – 65Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Optional replenishment systems Optimal review, min-max, or (s,S) system, like the P
system Reviews IP at fixed time intervals and places a variable-
sized order to cover expected needs Ensures that a reasonable large order is placed
Hybrid systems
Base-stock system Replenishment order is issued each time a withdrawal is
made Order quantities vary to keep the inventory position at R Minimizes cycle inventory, but increases ordering costs Appropriate for expensive items
12 – 66Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Quantity Discounts
Price incentives to purchase large quantities create pressure to maintain a large inventory
Item’s price is no longer fixed If the order quantity is increased enough, then the
price per unit is discounted A new approach is needed to find the best lot size
that balances: Advantages of lower prices for purchased materials and fewer
orders Disadvantages of the increased cost of holding more inventory
12 – 67Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Quantity Discounts
where P = price per unit
Total annual cost = Annual holding cost + Annual ordering or setup cost + Annual cost of materials
PDSQD
HQ
C 2
12 – 68Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Quantity Discounts
Unit holding cost (H) is usually expressed as a percentage of unit price
The lower the unit price (P) is, the lower the unit holding cost (H) is
The total cost equation yields U-shape total cost curves There are cost curves for each price level The feasible total cost begins with the top curve, then drops down,
curve by curve, at the price breaks EOQs do not necessarily produce the best lot size
The EOQ at a particular price level may not be feasible The EOQ at a particular price level may be feasible but may not
be the best lot size
12 – 69Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Two-Step Solution Procedure
Step 1. Beginning with lowest price, calculate the EOQ for each price level until a feasible EOQ is found. It is feasible if it lies in the range corresponding to its price. Each subsequent EOQ is smaller than the previous one, because P, and thus H, gets larger and because the larger H is in the denominator of the EOQ formula.
Step 2. If the first feasible EOQ found is for the lowest price level, this quantity is the best lot size. Otherwise, calculate the total cost for the first feasible EOQ and for the larger price break quantity at each lower price level. The quantity with the lowest total cost is optimal.
12 – 70Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Quantity Discounts
(a) Total cost curves with purchased materials added
(b) EOQs and price break quantities
PD forP = $4.00 PD for
P = $3.50 PD forP = $3.00
EOQ 4.00
EOQ 3.50
EOQ 3.00
Tota
l co
st (
dolla
rs)
Purchase quantity (Q)0 100 200 300
First price break
Second price break
Tota
l co
st (
dolla
rs)
Purchase quantity (Q)0 100 200 300
First price break
Second price break
C for P = $4.00
C for P = $3.50
C for P = $3.00
Figure D.3 – Total Cost Curves with Quantity Discounts
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Find Q with Quantity Discounts
EXAMPLE D.2
A supplier for St. LeRoy Hospital has introduced quantity discounts to encourage larger order quantities of a special catheter. The price schedule is
Order Quantity Price per Unit
0 to 299 $60.00
300 to 499 $58.80
500 or more $57.00
The hospital estimates that its annual demand for this item is 936 units, its ordering cost is $45.00 per order, and its annual holding cost is 25 percent of the catheter’s unit price. What quantity of this catheter should the hospital order to minimize total costs? Suppose the price for quantities between 300 and 499 is reduced to $58.00. Should the order quantity change?
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Find Q with Quantity Discounts
SOLUTION
Step 1: Find the first feasible EOQ, starting with the lowest price level:
HDS2
EOQ 0057.
units 77
005725000459362
.$..$
A 77-unit order actually costs $60.00 per unit, instead of the $57.00 per unit used in the EOQ calculation, so this EOQ is infeasible. Now try the $58.80 level:
HDS2
EOQ 8058.
units 76
805825000459362
.$..$
This quantity also is infeasible because a 76-unit order is too small to qualify for the $58.80 price. Try the highest price level:
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Find Q with Quantity Discounts
This quantity is feasible because it lies in the range corresponding to its price, P = $60.00
HDS2
EOQ 0060.
units 75
006025000459362
.$..$
Step 2: The first feasible EOQ of 75 does not correspond to the lowest price level. Hence, we must compare its total cost with the price break quantities (300 and 500 units) at the lower price levels ($58.80 and $57.00):
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Find Q with Quantity Discounts
PDSQD
HQ
C 2
284579360060004575936
00602502
7575 ,$.$.$.$. C
3825793680580045300936
80582502
300300 ,$.$.$.$. C
9995693600570045500936
00572502
500500 ,$.$.$.$. C
The best purchase quantity is 500 units, which qualifies for the deepest discount
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One-Period Decisions
Seasonal goods are a dilemma facing many retailers.
Newsboy problem
Step 1: List different demand levels and probabilities.
Step 2: Develop a payoff table that shows the profit for each purchase quantity, Q, at each assumed demand level, D.Each row represents a different order quantity and each column represents a different demand.The payoff depends on whether all units are sold at the regular profit margin which results in two possible cases.
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One-Period Decisions
If demand is high enough (Q ≤ D), then all of the cases are sold at the full profit margin, p, during the regular season
If the purchase quantity exceeds the eventual demand (Q > D), only D units are sold at the full profit margin, and the remaining units purchased must be disposed of at a loss, l, after the season
Payoff = (Profit per unit)(Purchase quantity) = pQ
Payoff = –(Demand)Lossperunit
Profit perunit soldduringseason
Amountdisposedof afterseason
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One-Period Decisions
Step 3: Calculate the expected payoff of each Q by using the expected value decision rule. For a specific Q, first multiply each payoff by its demand probability, and then add the products.
Step 4: Choose the order quantity Q with the highest expected payoff.
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Finding Q for One-Period Decisions
EXAMPLE D.3
One of many items sold at a museum of natural history is a Christmas ornament carved from wood. The gift shop makes a $10 profit per unit sold during the season, but it takes a $5 loss per unit after the season is over. The following discrete probability distribution for the season’s demand has been identified:
Demand 10 20 30 40 50
Demand Probability 0.2 0.3 0.3 0.1 0.1
How many ornaments should the museum’s buyer order?
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Finding Q for One-Period Decisions
SOLUTION
Each demand level is a candidate for best order quantity, so the payoff table should have five rows. For the first row, where Q = 10, demand is at least as great as the purchase quantity. Thus, all five payoffs in this row are
This formula can be used in other rows but only for those quantity–demand combinations where all units are sold during the season. These combinations lie in the upper-right portion of the payoff table, where Q ≤ D. For example, the payoff when Q = 40 and D = 50 is
Payoff = pQ = ($10)(10) = $100
Payoff = pQ = ($10)(40) = $400
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Finding Q for One-Period Decisions
The payoffs in the lower-left portion of the table represent quantity–demand combinations where some units must be disposed of after the season (Q > D). For this case, the payoff must be calculated with the second formula. For example, when Q = 40 and D = 30,
Using OM Explorer, we obtain the payoff table in Figure D.5
Payoff = pD – l(Q – D) = ($10)(30) – ($5)(40 – 30) = $250
Now we calculate the expected payoff for each Q by multiplying the payoff for each demand quantity by the probability of that demand and then adding the results. For example, for Q = 30,
Payoff = 0.2($0) + 0.3($150) + 0.3($300) + 0.1($300) + 0.1($300)= $195
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