CHAPTER 4 INVENTORY MANAGEMENT. LEARNING OBJECTIVES Define inventory and functions of inventory...
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Transcript of CHAPTER 4 INVENTORY MANAGEMENT. LEARNING OBJECTIVES Define inventory and functions of inventory...
CHAPTER 4CHAPTER 4
INVENTORY INVENTORY MANAGEMENTMANAGEMENT
LEARNING OBJECTIVESLEARNING OBJECTIVES
Define inventory and functions of inventory
Conduct an ABC analysis.
Explain and apply the EOQ and POQ model to solve typical problems.
Compute a ROP and safety stock
Inventory
A stock or store of goodsA stock of item kept to meet demand
Inventory Management How much
When
Classified
Accuracy
Objective of Inventory Control
To achieve satisfactory levels of customer service while keeping inventory costs within reasonable bounds
To keep enough inventory to meet customer demand and also be cost effective. Level of customer service
Costs of ordering and carrying inventory
Functions of Inventory
To meet anticipated demand
To decouple operations
To protect against stock-outs
To take advantage of order cycles
To help hedge against price increases
To take advantage of quantity discounts
Types of Inventory
Types of Inventory
Raw material Purchased but not processed
Work-in-process Undergone some change but not
completed A function of cycle time for a product
Maintenance/repair/operating (MRO) Necessary to keep machinery and
processes productive
Types of Inventory
Finished-goods inventories (manufacturing firms) or merchandise
(retail stores)Goods in transit
Completed product awaiting shipment
The Material Flow Cycle
InputInput Wait forWait for Wait toWait to MoveMove Wait in queueWait in queue SetupSetup RunRun OutputOutputinspectioninspection be movedbe moved timetime for operatorfor operator timetime timetime
Cycle timeCycle time
95%95% 5%5%
Effective Inventory Management
A system to keep track of inventory
A reliable forecast of demand
Knowledge of lead times
Reasonable estimates ofo Holding costs
o Ordering costs
o Shortage costs
A classification system
Inventory Counting Systems
Periodic Systemo Physical count of items made at periodic
intervals
Perpetual Inventory System o keeps track of removals from inventory
continuously, thus monitoring current levels of each item
Inventory Counting Systems
Two-Bin System Two containers of inventory; reorder when
the first is empty
Universal Bar Code printed on a label that has
information about the item to which it is attached
0
214800 232087768
Key Inventory TermsLead time: time interval between ordering and
receiving the order Item cost: Cost per item plus any other direct
costs associated with getting the item to the plant
Holding (carrying) costs: cost to carry an item in inventory for a length of time, usually a year
Ordering costs: costs of ordering and receiving inventory
Shortage costs: costs when demand exceeds supply
Independent Versus Dependent Demand Independent Demand
A
B(4) C(2)
D(2) E(1) D(3) F(2)
Dependent Demand
Independent demand is uncertain. Dependent demand is certain.
ABC Classification System
Classifying inventory according to some measure of importance and allocating control efforts accordingly.
AA - very important
BB - moderately important
CC - least important
Annual $ value of items
AA
BB
CC
High
Low
Low HighPercentage of Items
ABC Analysis
A ItemsA Items
B ItemsB ItemsC ItemsC Items
Pe
rce
nt
of
an
nu
al d
olla
r u
sa
ge
Pe
rce
nt
of
an
nu
al d
olla
r u
sa
ge
80 80 –
70 70 –
60 60 –
50 50 –
40 40 –
30 30 –
20 20 –
10 10 –
0 0 – | | | | | | | | | |
1010 2020 3030 4040 5050 6060 7070 8080 9090 100100
Percent of inventory itemsPercent of inventory items
ABC Analysis Example
Data Results - sorted by dollar volume
Dollar Volume Rank Item Volume Unit Cost Dollar volume
Dollar Volume Rank Item Volume Unit Cost Dollar Volume
Percent of Dollar Volume
Cum Percent of Dollar Volume Category
2 Item 1 1200 5.84 7,008.00$ 1 Item 4 1104 74.54 82,292.16$ 81.87% 81.87% A3 Item 2 1110 5.4 5,994.00$ 2 Item 1 1200 5.84 7,008.00$ 06.97% 88.84% B6 Item 3 896 1.12 1,003.52$ 3 Item 2 1110 5.4 5,994.00$ 05.96% 94.80% B1 Item 4 1104 74.54 82,292.16$ 4 Item 5 1110 2 2,220.00$ 02.21% 97.01% C4 Item 5 1110 2 2,220.00$ 5 Item 6 961 2.08 1,998.88$ 01.99% 99.00% C5 Item 6 961 2.08 1,998.88$ 6 Item 3 896 1.12 1,003.52$ 01.00% 100.00% C
Total 100,516.56$
Inventory Models for Independent Demand
Need to determine when and how much to order Basic economic order quantity Production order quantity Quantity discount model
Basic EOQ Model
Demand is known, constant, and independent
Lead time is known and constant Receipt of inventory is instantaneous and
complete Quantity discounts are not possible Only variable costs are setup and
holding Stockouts can be completely avoided
Cycle-Inventory Levels
Inventory depletion Inventory depletion (demand rate)(demand rate)
Receive Receive orderorder
1 cycle1 cycle
On
-han
d i
nve
nto
ry (
un
its)
On
-han
d i
nve
nto
ry (
un
its)
TimeTime
AverageAveragecyclecycleinventoryinventory
QQ——22
Total Annual Cycle-Inventory Costs
Objective is to minimize total costs
Table 11.5Table 11.5
An
nu
al c
ost
An
nu
al c
ost
Order quantityOrder quantity
Curve for total Curve for total cost of holding cost of holding
and setupand setup
Holding cost Holding cost curvecurve
Setup (or order) Setup (or order) cost curvecost curve
Minimum Minimum total costtotal cost
Optimal order Optimal order quantity (Q*)quantity (Q*)
Total cost = (Total cost = (HH) + () + (SS))DDQQ
QQ22
Holding cost = (Holding cost = (HH))QQ22
Ordering cost = (Ordering cost = (SS))DDQQ
Minimum Total Cost The total cost curve reaches its minimum where
the carrying and ordering costs are equal.
Q2
H DQ
S=
Cost Holding AnnualCost) Setupor der Demand)(Or 2(Annual
=
H
2DS = Q
Example
Bird feeder sales are 18 units per week, and the supplier charges $60 per unit. The cost of placing an order (S) with the supplier is $45.
Annual holding cost (H) is 25% of a feeder’s value, based on operations 52 weeks per year.
Management chose a 390-unit lot size (Q) so that new orders could be placed less frequently.
What is the annual cycle-inventory cost (C) of the current policy of using a 390-unit lot size?
Costing out a Lot Sizing Policy
What is the annual cycle-inventory cost (C) of the current policy of using a 390-unit lot size?
D = (18 /week)(52 weeks) = 936 units H = 0.25 ($60/unit) = $15
C = $2925 + $108 = $3033
C = (H) + (S) = (15) + (45) Q2
DQ
936390
3902
Museum of Natural History Gift Shop:
Would a lot size of 468 be better?
Lot Sizing at the Museumof Natural History Gift Shop
D = 936 units; H = $15; S = $45; Q = 390 units; C = $3033
C = (H) + (S) = (15) + (45) Q2
DQ
936468
4682
Q = 468 units; C = ?
C = $3510 + $90 = $3600
Q = 468 is a more expensive option.
The best lot size (EOQ) is the lowest point on the total annual cost curve!
3000 3000 —
2000 2000 —
1000 1000 —
0 0 —| | | | | | | |
5050 100100 150150 200200 250250 300300 350350 400400
Lot Size (Q)
Ann
ual c
ost
(dol
lars
)A
nnua
l cos
t (d
olla
rs) Total costTotal cost
Holding costHolding cost
Ordering costOrdering cost
Currentcost
CurrentQ
Lowestcost
Best Q (EOQ)
Lot Sizing at the Museumof Natural History Gift Shop
Computing the EOQ
C = (H) + (S)Q2
DQ
EOQ = 2DS
H
D = annual demandS = ordering or setup costs per lotH = holding costs per unit
D = 936 unitsH = $15S = $45
EOQ = 2(936)4515
= 74.94 or 75 units
C = (15) + (45)752
93675
C = $1,124.10
Bird Feeders:
Computing EOQ using the Excel Solver
= T == T =Expected Expected
time between time between ordersorders
Number of working Number of working days per yeardays per year
NN
= N = == N = =Expected Expected number of number of
ordersorders
DemandDemandOrder quantityOrder quantity
DDQ*Q*
TBOEOQ =EOQ
D
Understanding the Effect of Changes
A Change in the Demand Rate (D): When demand rises, the lot size also rises, but more slowly than actual demand.
A Change in the Setup Costs (S): Increasing S increases the EOQ and, consequently, the average cycle inventory.
A Change in the Holding Costs (H): EOQ declines when H increases.
EOQ: Robust Model ?
Errors in Estimating D, H, and S: Total cost is fairly insensitive to errors, even when the estimates are wrong by a large margin. The reasons are that errors tend to cancel each other out and that the square root reduces the effect of the error.
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.
Determinants of the Reorder Point
The rate of demandThe lead timeDemand and/or lead time variabilityStockout risk (safety stock)
Safety Stock
LT Time
Expected demandduring lead time
Maximum probable demandduring lead time
ROP
Qu
an
tity
Safety stockSafety stock reduces risk ofstockout during lead time
Reorder Point
ROP ROP ==Lead time for a Lead time for a
new order in daysnew order in daysDemand Demand per dayper day
== d x L d x L
d = d = DDNumber of working days in a yearNumber of working days in a year
Q*Q*
ROP ROP (units)(units)In
ven
tory
lev
el (
un
its)
Inve
nto
ry l
evel
(u
nit
s)
Time (days)Time (days)Lead time = LLead time = L
Slope = units/day = dSlope = units/day = d
Production Order Quantity Model
Used when inventory builds up over a Used when inventory builds up over a period of time after an order is placedperiod of time after an order is placed
Used when units are produced and sold Used when units are produced and sold simultaneouslysimultaneously
Inve
nto
ry l
evel
Inve
nto
ry l
evel
TimeTime
Demand part of cycle Demand part of cycle with no productionwith no production
Part of inventory cycle during Part of inventory cycle during which production (and usage) which production (and usage) is taking placeis taking place
tt
Maximum Maximum inventoryinventory
Production Order Quantity Model
Production Order Quantity Model
Q =Q = Number of pieces per orderNumber of pieces per order p = p = Daily production rateDaily production rateH =H = Holding cost per unit per yearHolding cost per unit per year d = d = Daily demand/usage rateDaily demand/usage ratet =t = Length of the production run in daysLength of the production run in days
= –= –Maximum Maximum inventory levelinventory level
Total produced during Total produced during the production runthe production run
Total used during Total used during the production runthe production run
== pt – dt pt – dt
H
2
IS
Q
DTC MAX
EPQ
= –= –Maximum Maximum inventory levelinventory level
Total produced during Total produced during the production runthe production run
Total used during Total used during the production runthe production run
== pt – dt pt – dt
However, Q = total produced = pt ; thus t = Q/pHowever, Q = total produced = pt ; thus t = Q/p
Maximum Maximum inventory levelinventory level = p – d = Q = p – d = Q 1 –1 –QQ
ppQQpp
ddpp
Holding cost = Holding cost = ((HH)) = = 1 –1 – H H ddpp
QQ22
Maximum inventory levelMaximum inventory level
22
Production Order Quantity Model
QQ22 = =22DSDS
HH[1 - ([1 - (dd//pp)])]
QQ* =* =22DSDS
HH[1 - [1 - ((dd//pp)])]
pp
Setup cost Setup cost == ((DD//QQ))SS
Holding cost Holding cost == HQ HQ[1 - ([1 - (dd//pp)])]1122
((DD//QQ))S = HQS = HQ[1 - ([1 - (dd//pp)])]1122
Production Order Quantity Model
EPQ.xls
Quantity Discount Models
Reduced prices are often available when larger quantities are purchased
Trade-off is between reduced product cost and increased holding cost
Total cost = Setup cost + Holding cost + Product costTotal cost = Setup cost + Holding cost + Product cost
TC = S + H + PDTC = S + H + PDDDQQ
QQ22
Total Cost With PDC
ost
EOQ/POQ
TC with PD
TC without PD
PD
0 Quantity
Adding Purchasing costdoesn’t change EOQ
Quantity Discount Models
Discount Number Discount Quantity Discount (%)
Discount Price (P)
1 0 to 999 no discount $5.00
2 1,000 to 1,999 4 $4.80
3 2,000 and over 5 $4.75
Table 12.2Table 12.2
A typical quantity discount scheduleA typical quantity discount schedule
1. For each discount, calculate Q*
2. If Q* for a discount doesn’t qualify, choose the smallest possible order size to get the discount
3. Compute the total cost for each Q* or adjusted value from Step 2
4. Select the Q* that gives the lowest total cost
Steps in analyzing a quantity discountSteps in analyzing a quantity discount
Quantity Discount Models
1,0001,000 2,0002,000
To
tal
cost
$T
ota
l co
st $
00
Order quantityOrder quantity
Q* for discount 2 is below the allowable range at point a Q* for discount 2 is below the allowable range at point a and must be adjusted upward to 1,000 units at point band must be adjusted upward to 1,000 units at point b
aabb
1st price 1st price breakbreak
2nd price 2nd price breakbreak
Total cost Total cost curve for curve for
discount 1discount 1
Total cost curve for discount 2Total cost curve for discount 2
Total cost curve for discount 3Total cost curve for discount 3
Figure 12.7Figure 12.7
Quantity Discount Models
Calculate Q* for every discountCalculate Q* for every discount Q* =2DSIP
QQ11* * = = 700= = 700 cars/order cars/order2(5,000)(49)2(5,000)(49)
(.2)(5.00)(.2)(5.00)
QQ22* * = = 714= = 714 cars/order cars/order2(5,000)(49)2(5,000)(49)
(.2)(4.80)(.2)(4.80)
QQ33* * = = 718= = 718 cars/order cars/order2(5,000)(49)2(5,000)(49)
(.2)(4.75)(.2)(4.75)
Quantity Discount Models
Calculate Q* for every discountCalculate Q* for every discount Q* =2DSIP
QQ11* * = = 700= = 700 cars/order cars/order2(5,000)(49)2(5,000)(49)
(.2)(5.00)(.2)(5.00)
QQ22* * = = 714= = 714 cars/order cars/order2(5,000)(49)2(5,000)(49)
(.2)(4.80)(.2)(4.80)
QQ33* * = = 718= = 718 cars/order cars/order2(5,000)(49)2(5,000)(49)
(.2)(4.75)(.2)(4.75)
1,0001,000 — adjusted — adjusted
2,0002,000 — adjusted — adjusted
Quantity Discount Models
Discount Number
Unit Price
Order Quantity
Annual Product
Cost
Annual Ordering
Cost
Annual Holding
Cost Total
1 $5.00 700 $25,000 $350 $350 $25,700
2 $4.80 1,000 $24,000 $245 $480 $24,725
3 $4.75 2,000 $23.750 $122.50 $950 $24,822.50
Table 12.3Table 12.3
Choose the price and quantity that gives Choose the price and quantity that gives the lowest total costthe lowest total cost
Buy Buy 1,0001,000 units at units at $4.80$4.80 per unit per unit
Quantity Discount Models
Quantity Discount Example: Collin’s Sport store is considering going to a different hat supplier. The present supplier charges $10 each and requires minimum quantities of 490 hats. The annual demand is 12,000 hats, the ordering cost is $20, and the inventory carrying cost is 20% of the hat cost, a new supplier is offering hats at $9 in lots of 4000. Who should he buy from?
Since the EOQ of 516 is not feasible, calculate the total cost (C) for each price to make the decision
4000 hats at $9 each saves $19,320 annually. Space?
$101,66012,000$9$1.802
4000$20
4000
12,000C
$120,98012,000$10$22
490$20
490
12,000C
$9
$10
Probabilistic Models and Safety Stock
Used when demand is not constant or certain
Use safety stock to achieve a desired service level and avoid stockouts
ROP ROP == d x L d x L + + ssss
Annual stockout costs = the sum of the units short x the probability x the stockout cost/unit
x the number of orders per year
Safety Stock Example
Number of Units Probability
30 .2
40 .2
ROP 50 .3
60 .2
70 .1
1.0
ROP ROP = 50= 50 units units Stockout cost Stockout cost = $40= $40 per frame per frameOrders per year Orders per year = 6= 6 Carrying cost Carrying cost = $5= $5 per frame per year per frame per year
ROP ROP = 50= 50 units units Stockout cost Stockout cost = $40= $40 per frame per frameOrders per year Orders per year = 6= 6 Carrying cost Carrying cost = $5= $5 per frame per year per frame per year
Safety Stock
Additional Holding Cost Stockout Cost
Total Cost
20 (20)($5) = $100 $0 $100
10 (10)($5) = $ 50 (10)(.1)($40)(6) = $240 $290
0 $ 0 (10)(.2)($40)(6) + (20)(.1)($40)(6) = $960 $960
A safety stock of A safety stock of 2020 frames gives the lowest total cost frames gives the lowest total cost
ROP ROP = 50 + 20 = 70= 50 + 20 = 70 frames frames
Safety Stock Example
Safety stock 16.5 units
ROP ROP
Place Place orderorder
Probabilistic DemandIn
ven
tory
lev
elIn
ven
tory
lev
el
TimeTime00
Minimum demand during lead timeMinimum demand during lead time
Maximum demand during lead timeMaximum demand during lead time
Mean demand during lead timeMean demand during lead time
Normal distribution probability of Normal distribution probability of demand during lead timedemand during lead time
Expected demand during lead time Expected demand during lead time (350(350 kits kits))
ROP ROP = 350 += 350 + safety stock of safety stock of 16.5 = 366.516.5 = 366.5
Receive Receive orderorder
Lead Lead timetime
Figure 12.8Figure 12.8
Safety Safety stockstock
Probability ofProbability ofno stockoutno stockout
95% of the time95% of the time
Mean Mean demand demand
350350
ROP = ? kitsROP = ? kits QuantityQuantity
Number of Number of standard deviationsstandard deviations
00 zz
Risk of a stockout Risk of a stockout (5% of area of (5% of area of normal curve)normal curve)
Probabilistic Demand
Use prescribed service levels to set safety stock when the cost of stockouts cannot be determined
ROP = demand during lead time + ZROP = demand during lead time + ZdLTdLT
where Z =number of standard deviations
dLT =standard deviation of demand during lead time
Probabilistic Demand
Average demand = = 350 kitsStandard deviation of demand during lead time = dLT = 10 kits5% stockout policy (service level = 95%)
Using Appendix I, for an area under the curve of 95%, the Z = 1.65
Safety stock Safety stock == Z ZdLTdLT = 1.65(10) = 16.5= 1.65(10) = 16.5 kits kits
Reorder point =expected demand during lead time + safety stock=350 kits + 16.5 kits of safety stock=366.5 or 367 kits
Probabilistic Example
Other Probabilistic Models
1. When demand is variable and lead time is constant
2. When lead time is variable and demand is constant
3. When both demand and lead time are variable
When data on demand during lead time is not available, there are other models available
Demand is variable and lead time is constantDemand is variable and lead time is constant
ROP ROP == ((average daily demand average daily demand x lead time in daysx lead time in days) +) + Z ZdLTdLT
wherewhere dd == standard deviation of demand per day standard deviation of demand per day
dLTdLT = = dd lead timelead time
Other Probabilistic Models
Variance = daily variance x no. of days of lead time
Standard D . (sum of daily variance during lead time)LL
L
dd
d
2
2
Average daily demand (normally distributed) = 15Standard deviation = 5Lead time is constant at 2 days90% service level desired
Z for Z for 90%90% = 1.28= 1.28From Appendix IFrom Appendix I
ROPROP = (15 = (15 units x units x 22 days days) +) + Z Zdltdlt
= 30 + 1.28(5)( 2)= 30 + 1.28(5)( 2)
= 30 + 9.02 = 39.02 ≈ 39= 30 + 9.02 = 39.02 ≈ 39
Safety stock is about 9 iPods
Probabilistic Example
Lead time is variable and demand is constantLead time is variable and demand is constant
ROP ROP ==((daily demand x average daily demand x average lead time in dayslead time in days) + ) + Z xZ x ( (daily daily demanddemand) ) xx LTLT
wherewhere LTLT == standard deviation of lead time in days standard deviation of lead time in days
Other Probabilistic Models
Probabilistic Example
Daily demand (constant) = 10Average lead time = 6 daysStandard deviation of lead time = LT = 398% service level desired
Z for Z for 98%98% = 2.055= 2.055From Appendix IFrom Appendix I
ROPROP = (10 = (10 units x units x 66 days days) + 2.055(10) + 2.055(10 units units)(3))(3)
= 60 + 61.65 = 121.65= 60 + 61.65 = 121.65
Reorder point is about 122 cameras
Both demand and lead time are variableBoth demand and lead time are variable
ROP ROP == ((average daily demand average daily demand x average lead timex average lead time) +) + Z ZdLTdLT
where d = standard deviation of demand per day
LT = standard deviation of lead time in days
dLT = (average lead time x d2)
+ (average daily demand)2 x LT2
Other Probabilistic Models
Probabilistic Example
Average daily demand (normally distributed) = 150Standard deviation = d = 16Average lead time 5 days (normally distributed)Standard deviation = LT = 1 day95% service level desired Z for Z for 95%95% = 1.65= 1.65
From Appendix IFrom Appendix I
ROPROP = (150 = (150 packs x packs x 55 days days) + 1.65) + 1.65dLTdLT
= (150 x 5) + 1.65 (5 days x 16= (150 x 5) + 1.65 (5 days x 1622) + (150) + (15022 x 1 x 122))
= 750 + 1.65(154) = 1,004 = 750 + 1.65(154) = 1,004 packspacks
Total Q System Costs
Total cost = Annual Holding Cost +
Annual setup/ordering Cost +
Annual safety stock holding cost
dLTHs
QD
HQ
TC 2
Fixed-Period (P) Systems
Orders placed at the end of a fixed period
Inventory counted only at end of period
Order brings inventory up to target level
Only relevant costs are ordering and holding
Lead times are known and constant
Items are independent from one another
On
-han
d i
nve
nto
ryO
n-h
and
in
ven
tory
TimeTime
QQ11
QQ22
Target quantity Target quantity ((TT))
PP
QQ33
QQ44
PP
PP
Figure 12.9Figure 12.9
Fixed-Period (P) Systems
Order amount Order amount ((QQ)) = Target = Target ((TT)) - On- - On-hand inventory - Earlier orders not yet hand inventory - Earlier orders not yet
received (SR)+ Back ordersreceived (SR)+ Back orders
Q = 50 - 0 - 0 + 3 = 53 jackets
3 jackets are back ordered No jackets are in stockIt is time to place an order Target value = 50
Fixed-Period (P) Example
Periodic Review Systems: Calculations for TI
Targeted Inventory level:
TI = d(p + LT) + SS
d = average period demand
p = order interval (days, wks)
LT = lead time (days, wks)
SS = zσd
Replenishment Quantity (Q)=TI-OH
TLp
Periodic Review Systems Example
The KVS Pharmacy stocks a popular brand of over-the-counter flu and cold medicine. The average demand for the medicine is 6 packages per day, with a standard deviation of 1.2 packages. A vendor for the pharmaceutical company checks KVS’s stock every 60 days. During one visit the store had 8 packages in stock. The lead time to receive an order is 5 days. Determine the order size for this order period that will enable KVS to maintain a 95% service level.
Q = d(p + LT) + zσd - OH
= 6(60 + 5) + 1.65(1.2) - 8
= 397.96
TLp
560
Total P System Costs
Same three cost element as Q systemOrder quantity, Q will be the average
consumption of inventory during the p periods between order; Q =dP
TLpdZHS
dPD
HdP
TC
2
Q System Example
dLT = d LT = 5 2 = 7.1
Safety stock = zdLT = 1.28(7.1) = 9.1 or 9 units
Reorder point = dL + safety stock = 2(18) + 9 = 45 units
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% cycle-service level. What is the total cost of the Q system?
C = ($15) + ($45) + 9($15)75
2
936
75
C = $562.50 + $561.60 + $135 = $1259.10
P System ExampleBird feeder demand is normally distributed with a mean of 18 units per week and a standard deviation in weekly demand of 5 units, operating 52 weeks a year. Lead time (L) is 2 weeks and EOQ is 75 units with a safety stock of 9 units and a cycle-service level of 90%. Annual demand (D) is 936 units. What is the equivalent P system and total cost?
P = (52) = (52) = 4.2 or 4 weeksEOQ
D75
936Time between reviews =
d(P+LT) = P + LT = 5 6 = 12 units Standard deviation of demand over the protection period
T = Average demand during the protection interval + Safety stock
= d (P + LT) + zd(P + LT)
= (18 units/week)(6 weeks) + 1.28(12 units) = 123 units
© 2007 Pearson Education
D = (18 units/week)(52 weeks) = 936 units Safety Stock during P = 15 Holding Costs = $15/unit Ordering Costs = $45
d = 18 units L = 2 weeks Cycle/service level = 90% EOQ = 75 units
The time between reviews (P) = 4 weeks Average demand during P + Safety stock = T = 123 units
C = ($15) + ($45) + 15($15) 4(18)
2936
4(18)
C = $540 + $585 + $225 = $1350
P System Example continued
The total P-system cost for the bird feeders is:
The P system requires 15 units in safety stock, while the Q system only needs 9 units. If cost were the only criterion, the Q system would be the choice.
Inventory is only counted at each review period
May be scheduled at convenient times
May require only periodic checks of inventory levels
May result in stockouts between periods
May require increased safety stock
Fixed-Period (P) Systems
Comparison of Q and P SystemsComparison of Q and P Systems
P Systems
Convenient to administer Orders for multiple items from the same supplier
may be combined Inventory Position (IP) only required at review
Systems in which inventory records are always current are called Perpetual Inventory Systems
Review frequencies can be tailored to each item Possible quantity discounts Lower, less-expensive safety stocks
Q Systems
Single Period Inventory Model
The SPI model is designed for products that share the following characteristics: Sold at their regular price only during a single-time period Demand is highly variable but follows a known probability
distribution Salvage value is less than its original cost so money is lost when
these products are sold for their salvage value
Objective is to balance the gross profit of the sale of a unit with the cost incurred when a unit is sold after its primary selling period
SPI Model Example: Tee shirts are purchase in multiples of 10 for a charity event for $8 each. When sold during the event the selling price is $20. After the event their salvage value is just $2. From past events the organizers know the probability of selling different quantities of tee shirts within a range from 80 to 120
Payoff TableProb. Of Occurrence .20 .25 .30 .15 .10Customer Demand 80 90 100 110 120# of Shirts Ordered Profit
80 $960 $960 $960 $960 $960 $96090 $900 $1080 $1080 $1080 $1080 $1040
Buy 100 $840 $1020 $1200 $1200 $1200 $1083 110 $780 $ 960 $1140 $1320 $1320 $1068 120 $720 $ 900 $1080 $1260 $1440 $1026
Sample calculations:Payoff (Buy 110)= sell 100($20-$8) –((110-100) x ($8-$2))= $1140Expected Profit (Buy 100)= ($840 X .20)+($1020 x .25)+($1200 x .30) + ($1200 x .15)+($1200 x .10) = $1083