Date post: | 29-Mar-2015 |
Category: |
Documents |
Upload: | holden-bryne |
View: | 218 times |
Download: | 1 times |
1
Inventory Management and
Control
2
Inventory Defined
• Inventory is the stock of any item or resource held to meet future demand and can include: raw materials, finished products, component parts, supplies, and work-in-process
3
Inventory
Process stage
Demand Type
Number & Value Other
Raw Material WIP
Finished Goods
Independent Dependent
A Items B Items C Items
Maintenance Operating
Inventory Classifications
4
E(1)
Independent vs. Dependent Demand
B(4)
E(2)D(1)
C(2)
E(3)B(1)
A
Independent Demand (Demand for the final end-product or demand not related to other items; demand created by
external customers)
Dependent Demand
(Derived demand for component
parts, subassemblies,
raw materials, etc- used to produce final products)
Finishedproduct
Component parts
Independent demand is uncertain Dependent demand is certain
5
Inventory Models
• Independent demand – finished goods, items that are ready to be sold– E.g. a computer
• Dependent demand – components of finished products– E.g. parts that make up the computer
6
Types of Inventories (1 of 2)
• Raw materials & purchased parts
• Partially completed goods called work in progress
• Finished-goods inventories (manufacturing firms) or merchandise (retail stores)
7
Types of Inventories (2 of 2)
• Replacement parts, tools, & supplies
• Goods-in-transit to warehouses or customers
8
The Material Flow Cycle (1 of 2)
9
Run time: Job is at machine and being worked onSetup time: Job is at the work station, and the work station is
being "setup."Queue time: Job is where it should be, but is not being
processed because other work precedes it.Move time: The time a job spends in transitWait time: When one process is finished, but the job is waiting
to be moved to the next work area.
The Material Flow Cycle (2 of 2)
WaitTime
MoveTime
QueueTime
SetupTime
RunTimeInput
Cycle Time
Output
10
Performance Measures
• Inventory turnover (the ratio of annual cost of goods sold to average inventory investment)
• Days of inventory on hand (expected number of days of sales that can be supplied from existing inventory)
11
Functions of Inventory (1 of 2)
1. To “decouple” or separate various parts of the production process, ie. to maintain independence of operations
2. To meet unexpected demand & to provide high levels of customer service
3. To smooth production requirements by meeting seasonal or cyclical variations in demand
4. To protect against stock-outs
12
Functions of Inventory (2 of 2)
5. To provide a safeguard for variation in raw material delivery time
6. To provide a stock of goods that will provide a “selection” for customers
7. To take advantage of economic purchase-order size
8. To take advantage of quantity discounts
9. To hedge against price increases
13
• Higher costs– Item cost (if purchased)– Holding (or carrying) cost
• Difficult to control
• Hides production problems
• May decrease flexibility
Disadvantages of Inventory
14
Inventory Costs Holding (or carrying) costs
Costs for storage, handling, insurance, etc
Setup (or production change) costs Costs to prepare a machine or process for
manufacturing an order, eg. arranging specific equipment setups, etc
Ordering costs (costs of replenishing inventory) Costs of placing an order and receiving goods
Shortage costs Costs incurred when demand exceeds supply
15
Holding (Carrying) Costs
• Obsolescence• Insurance• Extra staffing• Interest• Pilferage• Damage• Warehousing• Etc.
16
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%
17
Ordering Costs
• Supplies
• Forms
• Order processing
• Clerical support
• etc.
18
Setup Costs
• Clean-up costs
• Re-tooling costs
• Adjustment costs
• etc.
19
Shortage Costs
• Backordering cost
• Cost of lost sales
20
Inventory Control System Defined
An inventory system is the set of policies and controls that monitor levels of inventory and determine what levels should be maintained, when stock should be replenished and how large orders should be
Answers questions as: When to order? How much to order?
21
Objective of Inventory Control
To achieve satisfactory levels of customer service while keeping inventory costs within reasonable bounds
Improve the Level of customer service
Reduce the Costs of ordering and carrying inventory
22
A system to keep track of inventory
A reliable forecast of demand
Knowledge of lead times
Reasonable estimates of Holding costs
Ordering costs
Shortage costs
A classification system
Requirements of an Effective Inventory Management
23
Inventory Counting (Control) Systems• Periodic System
Physical count of items made at periodic intervals; order is placed for a variable amount after fixed passage of time.
• Perpetual (Continuous) Inventory System System that keeps track of removals from inventory continuously, thus monitoring current levels of each item (constant amount is ordered when inventory declines to a predetermined level)
24
Inventory ModelsSingle-Period Inventory Model
One time purchasing decision (Examples: selling t-shirts at a football game, newspapers, fresh bakery products, fresh flowers)
Seeks to balance the costs of inventory over stock and under stock
Multi-Period Inventory Models Fixed-Order Quantity Models
• Event triggered (Example: running out of stock) Fixed-Time Period Models
• Time triggered (Example: Monthly sales call by sales representative)
25
Single-Period Inventory Model
26
Single-Period Inventory Model
• In a single-period model, items are received in the beginning of a period and sold during the same period. The unsold items are not carried over to the next period.
• The unsold items may be a total waste, or sold at a reduced price, or returned to the producer at some price less than the original purchase price.
• The revenue generated by the unsold items is called the salvage value.
27
•Single period model: It is used to handle ordering of perishables (fresh fruits, flowers) and other items with limited useful lives (newspapers, spare parts for specialized equipment).
3
(Newsboy Problem)
28
Shortage cost (Cost of Understocking)
• Shortage cost: generally, this cost represents unrealized profit per unit
(Cu=Revenue per unit – Cost per unit)
• If a shortage or stockout cost relates to a spare part for a machine, then shortage cost refers to the actual cost of lost production.
29
Excess cost (Cost of Over Stocking)
• Excess cost (Ce): difference between purchase cost and salvage value of items left over at the end of a period.
• If there is a cost associated with disposing of excess items, the salvage cost will be negative.
30
Single Period Model
Given the costs of overestimating/underestimating demand and the probabilities of various demand sizes the goal is to identify the order quantity or stocking level that will minimize the long-run excess (overstock)or shortage costs (understock).
31
Demand may be discrete or continuous. The demand of computer, newspaper, etc. is usually an integer. Such a demand is discrete. On the other hand, the demand of gasoline is not restricted to integers. Such a demand is continuous. Often, the demand of perishable food items such as fish or meat may also be continuous. •Consider an order quantity Q•Let p = probability (demand<Q)
= probability of not selling the Qth item.•So, (1-p) = probability of selling the Qth item.
Single-Period Models (Demand Distribution)
32
• Expected loss from the Qth item = • Expected profit from the Qth item = • So, the Qth item should be ordered if
• Decision Rule (Discrete Demand):– Order maximum quantity Q such that
where p = probability (demand<Q)
se
e
cc
cpor
cppc
s
s
,
)1(
epc
scp1
se
s
cc
cp
Single-Period Models (Discrete Demand)
33
Single-Period Model
es
s
CC
CP
es
s
CC
CP
sold be unit will y that theProbabilit
estimatedunder demand ofunit per Cost C
estimatedover demand ofunit per Cost C
:Where
s
e
P
The service level is the probability that demand will not exceed the stocking level. The service level determines the amount of stocking level to keep.
The service level is the probability that demand will not exceed the stocking level. The service level determines the amount of stocking level to keep.
34
Optimal Stocking Level (Choosing optimum Stocking level to minimize these costs is similar to
balancing a seesaw)
Service Level
So
Quantity
Ce Cs
Balance point
Service level =Cs
Cs + CeCs = Shortage cost per unitCe = Excess cost per unit
35
Service Level
Another way to define ‘Service Level’ is:
• proportion of cycles in which no stock-out occurs
36
Service Level
Order Cycle Demand Stock-Outs
1 180 02 75 03 235 454 140 05 180 06 200 107 150 08 90 09 160 010 40 0
Total 1450 55
Since there are two cycles out of ten in which a stockout occurs, service level is 80%. This translates to a 96% fill rate. There are a total of 1,450 units demand and 55 stockouts (which means that 1,395 units of demand are satisfied).
37
Single Period Model (Demand is represented by a discrete distribution)
• Unlike the continuous case where the optimal solution is found by determining So which makes the distribution function equal to the critical ratio cs / (cs + ce), in the discrete case, the critical ratio takes place between two values of F(So) or F(Q)
• The optimal So or Q corresponds to the higher value of F( So) or F(Q).
(Note that, in the discrete case, the distribution function increases by jumps)
SEE EXAMPLES 17 & 18 on page 576
38
Example : Demand for cookies: Demand Probability of Demand
1,800 dozen 0.052,000 0.102,200 0.202,400 0.302,600 0.202,800 0.103,000 0,05
Selling price=$0.69, cost=$0.49, salvage value=$0.29
What is the optimal number of cookies to make?
c
Single-Period Models (Discrete Demand)
39
Cs= 0.69-0.49=$0.2, Ce= 0.49-0.29=$0.2
Order maximum quantity, Q such that
Demand, Q Probability(demand) Probability(demand<Q), p
1,800 dozen 0.05 0.052,000 0.10 0.152,200 0.20 0.352,400 0.30 0.652,600 0.20 0.852,800 0.10 0.953,000 0,05 1.00
5.0demandyProbabilit
es cc
cQp
s
Single-Period Models (Discrete Demand)
40
Single-Period Models (Continous Demand)
• Often the demand is continuous. Even when the demand is not continuous, continuous distribution may be used because the discrete distribution may be inconvenient.
• We shall discuss two distributions:
Uniform distribution
Normal distribution
41
Example 2: The J&B Card Shop sells calendars. The once-a-year order for each year’s calendar arrives in September. The calendars cost $1.50 and J&B sells them for $3 each. At the end of July, J&B reduces the calendar price to $1 and can sell all the surplus calendars at this price. How many calendars should J&B order if the September-to-July demand can be approximated by
a. uniform distribution between 150 and 850
Single-Period Models (Continuous Demand)
42
Overage cost
ce = Purchase price - Salvage value =1.5-1=$0.5
Underage cost
cs = Selling price - Purchase price =3-1.5=$1.5
Single-Period Models (Continuous Demand)
43
p =0.75
Now, find the Q so that p = probability(demand<Q) =0.75
Q* = a+p(b-a) =150+0.75(850-150)=675
se
s
cc
c
Area =
850150 Demand
Pro
babi
lity
Area=
Q*
Single-Period Models (Continuous Demand)
44
Example 3: The J&B Card Shop sells calendars. The once-a-year order for each year’s calendar arrives in September. The calendars cost $1.50 and J&B sells them for $3 each. At the end of July, J&B reduces the calendar price to $1 and can sell all the surplus calendars at this price. How many calendars should J&B order if the September-to-July demand can be approximated by
b. normal distribution with = 500 and =120.
Single-Period Models (Continuous Demand)
45
Solution to Example 3: ce =$0.50, cu =$1.50 (see Example 2)
p = = 0.750.501.50
1.50
se
s
cc
c
Single-Period Models (Continuous Demand)
46
Single-Period Models (Continuous Demand)
Now, find the Q so that p = 0.75
47
Single-Period Models (Continuous Demand
Q= So = mean + zσ
= 500 + .68(120)
= 582
48
Single Period Example 15 (pg. 574)Demand is uniformly distributed
• Ce = $0.20 per unit
• Cs = $0.60 per unit
• Service level = Cs/(Cs+Ce) = .6/(.6+.2)
• Service level = .75
• Opt. Stock.Level=S0=300+.75(500-300)= 450 liters
Service Level = 75%
Quantity
Ce Cs
Stockout risk = 1.00 – 0.75 = 0.25
49
49
Uniform Distribution[Continuous Dist’n]
• A random variable X is uniformly distributed on the interval (a,b), U(a,b), if its pdf and cdf are:
• Properties– P(x1 < X < x2) is proportional to the length of the interval [F(x2) –
F(x1) = (x2-x1)/(b-a)]
– E(X) = (a+b)/2 V(X) = (b-a)2/12
• U(0,1) provides the means to generate random numbers, from which random variates can be generated.
otherwise ,0
,1
)( bxaabxf
bx
bxaab
axax
xF
,1
,
,0
)(
50
50
Poisson Distribution[Discrete Dist’n]
• Poisson distribution describes many random processes quite well and is mathematically quite simple.– where > 0, pdf and cdf are:
– E(X) = = V(X)
otherwise ,0
,...1,0 ,!)( x
x
exp
x
x
i
i
i
exF
0 !)(
51
51
Normal Distribution [Continuous
Dist’n]
• A normally distributed random variable X has the pdf:
– Mean:
– Variance:
– Denoted as X ~ N(,2)
• Special properties: .– symmetric about .
– The maximum value of the pdf occurs at x = ; the mean and mode are equal.
xx
xf ,2
1exp
2
1)(
2
02
52
52
Normal Distribution [Continuous
Dist’n]
• Evaluating the distribution:– Use numerical methods (no closed form)
– Independent of and using the standard normal distribution:
Z ~ N(0,1)
– Transformation of variables: let Z = (X - ) / ,
z t dtez 2/2
2
1)( where,
)()(
2
1
)(
/)(
/)( 2/2
xx
x z
dzz
dze
xZPxXPxF
53
53
Normal Distribution [Continuous
Dist’n]
• Example: The time required to load an oceangoing vessel, X, is distributed as N(12,4)– The probability that the vessel is loaded in less than 10 hours:
• Using the symmetry property, (1) is the complement of (-1)
1587.0)1(2
1210)10(
F
54
Single Period Model(Demand is represented by a continous distribution)
Our college basketball team is playing in a tournament game this weekend. Based on our past experience we sell on average 2,400 shirts with a standard deviation of 350 and we can assume that demand for shirts is approximately normally distributed. We make $10 on every shirt we sell at the game, but lose $5 on every shirt not sold. What is the optimal stocking level for shirts?
So =mean + zσCs = $10 and Ce = $5; P ≤ $10 / ($10 + $5) = .667
Z.667 = .432 therefore we need 2,400 + .432(350) = 2,551 shirts
55
Multi-Period Inventory Models
Fixed-Order Quantity Models (Types of)Economic Order Quantity Model (EOQ)Economic Production Order Quantity (Economic Lot Size) Model (EPQ)Economic Order Quantity Model with Quantity Discounts
Fixed Time Period (Fixed Order Interval) Models
56
Fixed Order Quantity Models:Economic Order Quantity Model
57
Economic Order Quantity Model Assumptions (1 of 2):
• Demand for the product is known with certainty, it is constant and uniform throughout the period
• Lead time (time from ordering to receipt) is known and constant
• Price per unit of product is constant (no quantity discounts). So it is not included in the total cost.
• Inventory holding cost is based on average inventory
58
Economic Order Quantity Model Assumptions (2 of 2):
• Ordering or setup costs are constant
• All demands for the product will be satisfied (no backorders are allowed)
• No stockouts (shortages) are allowed
• The order quantity is received all at once. (Instantaneous receipt of material in a single lot)The goal is to calculate the order quantitiy that
minimizes total cost
59
Basic Fixed-Order Quantity Model and Reorder Point Behavior
R = Reorder pointQ = Economic order quantityL = Lead time
L L
Q QQ
R
Time
Numberof unitson hand(Inv. Level)
1. You receive an order quantity Q.
2. You start using them up over time. 3. When you reach down to
a level of inventory of R, you place your next Q sized order.
4. The cycle then repeats.
60
EOQ Model
Reorder Point
(ROP)
Time
Inventory LevelAverageInventory
(Q/2)
Lead Time
Order Quantity
(Q)
Demand rate
Order placed Order received
61
EOQ Cost Model: How Much to Order?
By adding the holding and ordering costs together, we determine the total cost curve, which in turn is used to find the optimal order quantity that minimizes total costs
Slope = 0Slope = 0
Total CostTotal Cost
Order Quantity, Order Quantity, QQ
Annual Annual cost ($)cost ($)
Minimum Minimum total costtotal cost
Optimal orderOptimal order QQoptopt
Carrying Cost =Carrying Cost =HHQQ
22
Ordering Cost =Ordering Cost =SSDD
62
• More units must be stored if more are ordered
Purchase OrderDescription Qty.Microwave 1
Order quantity
Purchase OrderDescription Qty.Microwave 1000
Order quantity
Why Holding Costs Increase?
63
Cost is spread over more units
Example: You need 1000 microwave ovens
Purchase OrderDescription Qty.Microwave 1
Purchase OrderDescription Qty.Microwave 1
Purchase OrderDescription Qty.Microwave 1
Purchase OrderDescription Qty.Microwave 1
1000 Order (Postage $ 0.33) 1 Order (Postage $330)
Order quantity
Purchase OrderDescription Qty.Microwave 1000
Why Ordering Costs Decrease ?
64
Basic Fixed-Order Quantity (EOQ) Model Formula
H 2
Q + S
Q
D + DC = TC H
2
Q + S
Q
D + DC = TC
Total Annual =Cost
AnnualPurchase
Cost
AnnualOrdering
Cost
AnnualHolding
Cost+ +
TC=Total annual costD =Annual demandC =Cost per unitQ =Order quantityS =Cost of placing an order or setup costR =Reorder pointL =Lead timeH=Annual holding and storage cost per unit of inventory
TC=Total annual costD =Annual demandC =Cost per unitQ =Order quantityS =Cost of placing an order or setup costR =Reorder pointL =Lead timeH=Annual holding and storage cost per unit of inventory
65
EOQ Cost Model
Annual ordering cost =Annual ordering cost =S S DD
AnnualAnnual carrying costcarrying cost = =HHQQ
22
Total cost = +Total cost = +S S DD
QQH H QQ
22
TC = +S D
Q
H Q
2
=- +S D
Q2
H
2TC
Q
0 =- +S D
Q2
H
2
Qopt =2SD
H
Deriving Qopt Proving equality of costs at optimal point
=S D
Q
H Q
2
Q2 =2S D
H
Qopt =2 S D
H
Using calculus, we take the first derivative of the total cost function with respect to Q, and set the derivative (slope) equal to zero, solving for the optimized (cost minimized) value of Qopt
66
Deriving the EOQ
Q = 2DS
H =
2(Annual D em and)(Order or Setup Cost)
Annual Holding CostOPTQ =
2DS
H =
2(Annual D em and)(Order or Setup Cost)
Annual Holding CostOPT
Reorder point, R = d L_
Reorder point, R = d L_
(constant) timeLead = L
(constant) demanddaily average = d_
We also need a reorder point to tell us when to place an order
We also need a reorder point to tell us when to place an order
1) How much to order?
2) When to order?
67
Optimal Order Quantity
Expected Number of Orders
Expected Time Between Orders Working Days / Year
Working Days / Year
= =× ×
= =
= =
=
= ×
Q*D SH
ND
Q*
TN
dD
ROP d L
2
EOQ Model Equations
68
EOQ Example 1 (1 of 3)
Annual Demand = 1,000 unitsDays per year considered in average daily demand = 365Cost to place an order = $10Holding cost per unit per year = $2.50Lead time = 7 daysCost per unit = $15
Given the information below, what are the EOQ and reorder point?
Given the information below, what are the EOQ and reorder point?
69
EOQ Example 1(2 of 3)
Q = 2DS
H =
2(1,000 )(10)
2.50 = 89.443 units or OPT 90 unitsQ =
2DS
H =
2(1,000 )(10)
2.50 = 89.443 units or OPT 90 units
d = 1,000 units / year
365 days / year = 2.74 units / dayd =
1,000 units / year
365 days / year = 2.74 units / day
Reorder point, R = d L = 2.74units / day (7days) = 19.18 or _
20 units Reorder point, R = d L = 2.74units / day (7days) = 19.18 or _
20 units
In summary, you place an optimal order of 90 units. In the course of using the units to meet demand, place the next order of 90 units when you only have 20 units left.
In summary, you place an optimal order of 90 units. In the course of using the units to meet demand, place the next order of 90 units when you only have 20 units left.
70
EOQ Example I(3 of 3)
TCTCminmin = =SSDD
QQHHQQ
22
TCTCminmin = = (10)(1,000)(10)(1,000)
9090((2,52,5)()(990)0)
22
TCTCminmin = $ = $ 111 111 + $ + $111111 = = 22 22 $$
Orders per year =Orders per year = DD//QQoptopt
== 1000/1000/9900
== 1111 orders/year orders/year
Order cycle timeOrder cycle time== 365/(365/(DD//QQoptopt))
== 336565//1111 == 33.133.1daysdays
++
+
71
EOQ Example 2(1 of 2)
Annual Demand = 10,000 unitsDays per year considered in average daily demand = 365Cost to place an order = $10Holding cost per unit per year = 10% of cost per unitLead time = 10 daysCost per unit = $15
Determine the economic order quantity and the reorder point given the following…
Determine the economic order quantity and the reorder point given the following…
72
EOQ Example 2(2 of 2)
Q =2DS
H=
2(10,000 )(10)
1.50= 365.148 units, or OPT 366 unitsQ =
2DS
H=
2(10,000 )(10)
1.50= 365.148 units, or OPT 366 units
d =10,000 units / year
365 days / year= 27.397 units / dayd =
10,000 units / year
365 days / year= 27.397 units / day
R = d L = 27.397 units / day (10 days) = 273.97 or _
274 unitsR = d L = 27.397 units / day (10 days) = 273.97 or _
274 units
Place an order for 366 units. When in the course of using the inventory you are left with only 274 units, place the next order of 366 units.
Place an order for 366 units. When in the course of using the inventory you are left with only 274 units, place the next order of 366 units.
73
EOQ Example 3
HH = $0.75 per yard = $0.75 per yard SS = $150 = $150 DD = 10,000 yards = 10,000 yards
QQoptopt = =22 S S DD
HH
QQoptopt = =2(150)(10,000)2(150)(10,000)
(0.75)(0.75)
QQoptopt = 2,000 yards = 2,000 yards
TCTCminmin = + = +S S DD
QQH H QQ
22
TCTCminmin = + = +((150)(10,000)150)(10,000)
2,0002,000(0.75)(2,000)(0.75)(2,000)
22
TCTCminmin = $750 + $750 = $1,500 = $750 + $750 = $1,500
Orders per year = D/Qopt
= 10,000/2,000
= 5 orders/year
Order cycle time =311 days/(Order cycle time =311 days/(DD//QQoptopt))
== 311/5311/5
== 62.2 store days62.2 store days
74
When to Reorder with EOQ Ordering ?• Reorder Point – is the level of inventory at which a
new order is placed
ROP = d . L
• 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 (probability that inventory available during the lead time will meet the demand) 1 - Probability of stockout
75
Reorder Point Example
Demand = 10,000 yards/year
Store open 311 days/year
Daily demand = 10,000 / 311 = 32.154 yards/day
Lead time = L = 10 days
R = dL = (32.154)(10) = 321.54 yards
76
Determinants of the Reorder Point
• The rate of demand
• The lead time
• Demand and/or lead time variability
• Stockout risk (safety stock)
77
Answer how much & when to order Allow demand and lead time to vary
Follows normal distribution Other EOQ assumptions apply
Consider service level & safety stock Service level = 1 - Probability of stockout Higher service level means more safety stock More safety stock means higher ROP
Probabilistic Models
78
Safety Stock
LT Time
Expected demandduring lead time
Maximum probable demandduring lead time
ROP
Qu
an
tity
Safety stock
Safety stock reduces risk ofstockout during lead time
79
Reorder Point With Variable Demand
Reorderpoint, R
Q
LTLT
TimeTimeLTLT
Inve
nto
ry le
vel
0
80
Reorder Point with a Safety Stock
Reorderpoint, R
LT
Time
LT
Inve
nto
ry le
vel
0
Safety Stock
81
Reorder Point With Variable Demand and Constant Lead Time
R = dL + zd Lwhere
d = average daily demandL = lead time
d = the standard deviation of daily demand
z = number of standard deviationscorresponding to the service levelprobability
zd L = safety stock
82
Reorder Point for Service Level
Probability of meeting demand during lead time = service level
Probability of a stockout
R
Safety stock
dLExpected Demand
zd L
The reorder point based on a normal distribution of LT demand
83
Reorder Point for Variable Demand (Example)
The carpet store wants a reorder point with a 95% service level and a 5% stockout probability
d = 30 yards per day, (demand is normally distributed) d= 5 yards per day
For a 95% service level, z = 1.65
R = dL + z d L
= 30(10) + (1.65)(5)( 10)
= 326.1 yards
Safety stock = z d L
= (1.65)(5)( 10)
= 26.1 yards
L= 10 days
84
Shortages and Service Levels
It is also important to specify:
1) Expected number of units short per order cycle
E(n) =E(z) σdLT
where E(z) is standardized number of units short obtained from Table 12.3, pg. 569.
2) Expected number of units short per year
E(N) =E (n) (D/Q)
3) Annual Service Level
SLannual = 1- E(N)/D that is percentage of demand filled directly from inventory, known also as FILL RATE.
85
85
Example 10 (pg. 568)– shortages and service levels
Suppose standard deviation of lead time demand is known to be 20 units. Lead time demand is approximately normal.
(a) For lead time service level of 90 percent, determine the expected number of units short for any other cycle.
(b) What lead time service level would imply an expected shortage of 2 units?
86
86
Answer – shortage and service levels
(a) For lead time service level of 90 percent, determine the expected number of units short for any other cycle.
σdLT= 20 units
lead time service level is 0.90 from z table (lead time), E(z)= 0.048 (page 569, table 12.3)
E(n) =E(z) σdLT = (0.048) (20)= 0.96 or about 1 unit.
(b) What lead time service level would an expected shortage of 2 units imply?
E(n) = 2 E(n) =E(z) σdLT or E(z) = E(n) / σdLT =(2)/(20)= 0.100 from the
table, lead time service level is 81.06 percent or 81.7%
87
Shortages and Service Levels
• Expected number of units short per year
See example 11, page 568
• Annual Service Level
See example 12, page 570
Note that annual service level will usually be grater than the cycle service level
88
Fixed Order Quantity Models:-Noninstantaneous Receipt-Production Order Quantity
(Economic Lot Size) Model
89
Production done in batches or lotsCapacity to produce a part exceeds that part’s
usage or demand rateAllows partial receipt of material
Other EOQ assumptions apply
Suited for production environment Material produced, used immediately Provides production lot size
Lower holding cost than EOQ modelAnswers how much to order and when to order
Production Order Quantity Model
90
POQ Model Inventory Levels (1 of 2)Inventory Level
TimeSupply Begins
Supply Ends
Production portion of cycle
Demand portion of cycle with no supply
Maximum inventory level
91
POQ Model Inventory Levels (2 of 2)
Time
Inventory Level
Production Portion of
Cycle
Max. Inventory Q/p·(p- u)Q*Q*
Supply Begins
Supply Ends
Inventory level with no demand
Demand portion of cycle with no supply
Average inventory Q/2(1- u/p)
92
D = Demand per year
S = Setup cost
H = Holding cost
d = Demand per day
p = Production per day
POQ Model Equations
Setup Cost
Holding Cost
= *
= *
=
Q
D
QS
(
1/2 * H * Q -u
p1
)-u
p1
( )
Maximum inventory level
93
Production Order Quantity Example (1 of 2)
H = $0.75 per yard S = $150 D = 10,000 yards
u = 10,000/311 = 32.2 yards per day p = 150 yards per day
POQopt = = = 2,256.8 yards
2 S D
H 1 - up
2(150)(10,000)
0.75 1 - - 32.2150
TC = + 1 - = $1,329up
S DQ
H Q2
Production run = = = 15.05 days per orderQp
2,256.8150
94
Production Quantity Example (2 of 2)
H = $0.75 per yard S = $150 D = 10,000 yards
u= 10,000/311 = 32.2 yards per day p = 150 yards per day
QQoptopt = = = 2,256.8 yards = = = 2,256.8 yards
22CCooDD
CCcc 1 - 1 - ddpp
2(150)(10,000)2(150)(10,000)
0.75 1 - 0.75 1 - 32.232.2150150
TCTC = + 1 - = $1,329 = + 1 - = $1,329ddpp
CCooDD
CCccQQ
22
Production run = = = 15.05 days per orderQp
2,256.8150
Number of production runs = = = 4.43 runs/yearDQ
10,0002,256.8
Maximum inventory level = Q 1 - = 2,256.8 1 -
= 1,772 yards
up
32.2150
95
Fixed-Order Quantity Models:Economic Order Quantity Model
with Quantity Discounts
96
• Answers how much to order & when to order
• Allows quantity discounts
– Price per unit decreases as order quantity increases
– Other EOQ assumptions apply
• Trade-off is between lower price & increased holding cost
Quantity Discount Model
TC = + + PDS D
Q
iP QQ
22Where P: Unit Price
Total cost with purchasing cost
97
Total Costs with Purchasing CostC
ost
EOQ
TC with PD
TC without PD
PD
0 Quantity
Adding Purchasing costdoesn’t change EOQ
98
Quantity Discount Models
• There are two general cases of quantity discount models:
1.Carrying costs are constant (e.g. $2 per unit).
2.Carrying costs are stated as a percentage off purchase price (20% of unit price)
99
1) Total Cost with Constant Carrying Costs(Compute the Common Optimal Order Quantity
OC
EOQ Quantity
Tot
al C
ost TCa
TCc
TCbDecreasing Price
CC a,b,c
1002) Total Cost with Variable Carrying Cost (Compute Optimal Order Quantity for each price
range)
Cost Holding Annual
Cost) Setupor der Demand)(Or 2(Annual =
iC
2DS = QOPT
Based on the same assumptions as the EOQ model, the price-break model has a similar Qopt formula:
i = percentage of unit cost attributed to carrying inventoryC = cost per unit
Since “C” changes for each price-break, the formula above will have to be used with each price-break cost value
101
Quantity Discount – How Much to Order?
102
Price-Break Example 1 (1 of 3)
ORDER SIZE PRICE
0 - 99 $10
100 - 199 8 (d1)
200+ 6 (d2)
For this problem holding cost is given as a constant value, not as a percentage of price, so the optimal order quantity is the same for each of the price ranges. (see the figure 12.9)
103
Price Break Example 1 (2 of 3)
QQoptopt
Carrying cost Carrying cost
Ordering cost Ordering cost
Inve
nto
ry c
ost
($)
Inve
nto
ry c
ost
($)
QQ((dd1 1 ) = 100) = 100 QQ((dd2 2 ) = 200) = 200
TC TC ((dd2 2 = $6 ) = $6 )
TCTC ( (dd1 1 = $8 )= $8 )
TC TC = ($10 )= ($10 )
104
Price Break Example 1 (3 of 3)
QQoptopt
Carrying cost Carrying cost
Ordering cost Ordering cost
Inve
nto
ry c
ost
($)
Inve
nto
ry c
ost
($)
QQ((dd1 1 ) = 100) = 100 QQ((dd2 2 ) = 200) = 200
TC TC ((dd2 2 = $6 ) = $6 )
TCTC ( (dd1 1 = $8 )= $8 )
TC TC = ($10 )= ($10 )
The lowest total cost is at the second price break
105
Price Break Example 2
QUANTITYQUANTITY PRICEPRICE
1 - 491 - 49 $1,400$1,400
50 - 8950 - 89 1,1001,100
90+90+ 900900
SS = = $2,500 $2,500
HH = = $190 per computer $190 per computer
DD = = 200200
QQoptopt = = = 72.5 PCs = = = 72.5 PCs22SSDD
HH2(2500)(200)2(2500)(200)
190190
TCTC = + + = + + PD PD = $233,784 = $233,784 SSDD
QQoptopt
H H QQoptopt
22
For For QQ = 72.5 = 72.5
TCTC = + + = + + PD PD = $194,105= $194,105SSDD
H H QQ
22
For For QQ = 90 = 90
106
Price-Break Example 3 (1 of 4)
A company has a chance to reduce their inventory ordering costs by placing larger quantity orders using the price-break order quantity schedule below. What should their optimal order quantity be if this company purchases this single inventory item with an e-mail ordering cost of $4, a carrying cost with a rate of 2% of the unit price, and an annual demand of 10,000 units?
A company has a chance to reduce their inventory ordering costs by placing larger quantity orders using the price-break order quantity schedule below. What should their optimal order quantity be if this company purchases this single inventory item with an e-mail ordering cost of $4, a carrying cost with a rate of 2% of the unit price, and an annual demand of 10,000 units?
Order Quantity(units) Price/unit($)0 to 2,499 $1.202,500 to 3,999 1.004,000 or more .98
107
Price-Break Example (2 of 4)
units 1,826 = 0.02(1.20)
4)2(10,000)( =
iC
2DS = QOPT
Annual Demand (D)= 10,000 unitsCost to place an order (S)= $4
First, plug data into formula for each price-break value of “C”
units 2,000 = 0.02(1.00)
4)2(10,000)( =
iC
2DS = QOPT
units 2,020 = 0.02(0.98)
4)2(10,000)( =
iC
2DS = QOPT
Carrying cost % of total cost (i)= 2%Cost per unit (C) = $1.20, $1.00, $0.98
Interval from 0 to 2499, the Qopt value is feasible
Interval from 2500-3999, the Qopt value is not feasible
Interval from 4000 & more, the Qopt value is not feasible
Next, determine if the computed Qopt values are feasible or not
108
Price-Break Example 2 (3 of 4)
Since the feasible solution occurred in the first price-break, it means that all the other true Qopt values occur at the beginnings of each price-break interval. Why?
Since the feasible solution occurred in the first price-break, it means that all the other true Qopt values occur at the beginnings of each price-break interval. Why?
0 1826 2500 4000 Order Quantity
Total annual costs
So the candidates for the price-breaks are 1826, 2500, and 4000 units
So the candidates for the price-breaks are 1826, 2500, and 4000 units
Because the total annual cost function is a “u” shaped function
Because the total annual cost function is a “u” shaped function
109
Price-Break Example 2 (4 of 4)
iC 2
Q + S
Q
D + DC = TC iC
2
Q + S
Q
D + DC = TC
Next, we plug the true Qopt values into the total cost annual cost function to determine the total cost under each price-break
Next, we plug the true Qopt values into the total cost annual cost function to determine the total cost under each price-break
TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20) = $12,043.82TC(2500-3999)= $10,041TC(4000&more)= $9,949.20
TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20) = $12,043.82TC(2500-3999)= $10,041TC(4000&more)= $9,949.20
Finally, we select the least costly Qopt, which in this problem occurs in the 4000 & more interval. In summary, our optimal order quantity is 4000 units
Finally, we select the least costly Qopt, which in this problem occurs in the 4000 & more interval. In summary, our optimal order quantity is 4000 units
110
Multi-period Inventory Models:Fixed Time Period
(Fixed-Order- Interval) Models
111
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
Fixed-Order-Interval Model
112
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
pp pp pp
QQ11 QQ22
QQ33
QQ44
Target maximum
TimeTime
d In
vent
ory
d In
vent
ory
113
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
Fixed-Interval Benefits
114
Requires a larger safety stock Increases carrying cost Costs of periodic reviews
Fixed-Interval Disadvantages
115
Fixed-Time Period Model with Safety Stock Formula
order)on items (includes levelinventory current = I
timelead and review over the demand ofdeviation standard =
yprobabilit service specified afor deviations standard ofnumber the= z
demanddaily averageforecast = d
daysin timelead = L
reviewsbetween days ofnumber the= T
ordered be toquantitiy = q
:Where
I - Z+ L)+(Td = q
L+T
L+T
order)on items (includes levelinventory current = I
timelead and review over the demand ofdeviation standard =
yprobabilit service specified afor deviations standard ofnumber the= z
demanddaily averageforecast = d
daysin timelead = L
reviewsbetween days ofnumber the= T
ordered be toquantitiy = q
:Where
I - Z+ L)+(Td = q
L+T
L+T
q = Average demand + Safety stock – Inventory currently on handq = Average demand + Safety stock – Inventory currently on hand
116
Fixed-Time Period Model: Determining the Value of T+L
T+L di 1
T+L
d
T+L d2
=
Since each day is independent and is constant,
= (T + L)
i
2
T+L di 1
T+L
d
T+L d2
=
Since each day is independent and is constant,
= (T + L)
i
2
The standard deviation of a sequence of random events equals the square root of the sum of the variances
117
Order Quantity for a Periodic Inventory System
Q = d(tb + L) + zd T + L - Iwhere
d = average demand rateT = the fixed time between ordersL = lead time
d = standard deviation of demand
zd T + L = safety stockI = inventory level
z = the number of standard deviations for a specified service level
118
Fixed-Period Model with Variable Demand (Example 1)
d = 6 bottles per dayd = 1.2 bottlesT = 60 daysL = 5 daysI = 8 bottlesz = 1.65 (for a 95% service level)
Q = d(T + L) + zd T + L - I
= (6)(60 + 5) + (1.65)(1.2) 60 + 5 - 8
= 397.96 bottles
119
Fixed-Time Period Model withVariable Demand (Example 2)(1 of 3)
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.
Given the information below, how many units should be ordered?
Given the information below, how many units should be ordered?
120
Fixed-Time Period Model with Variable Demand (Example 2)(2 of 3)
T+ L d2 2 = (T + L) = 30 + 10 4 = 25.298 T+ L d
2 2 = (T + L) = 30 + 10 4 = 25.298
So, by looking at the value from the Table, we have a probability of 0.9599, which is given by a z = 1.75
So, by looking at the value from the Table, we have a probability of 0.9599, which is given by a z = 1.75
121
Fixed-Time Period Model with Variable Demand (Example 2) (3 of 3)
or 644.272, = 200 - 44.272 800 = q
200- 298)(1.75)(25. + 10)+20(30 = q
I - Z+ L)+(Td = q L+T
units 645
or 644.272, = 200 - 44.272 800 = q
200- 298)(1.75)(25. + 10)+20(30 = q
I - Z+ L)+(Td = q L+T
units 645
So, to satisfy 96 percent of the demand, you should place an order of 645 units at this review period
So, to satisfy 96 percent of the demand, you should place an order of 645 units at this review period
122
ABC Classification System
• Demand volume and value of items vary
• Items kept in inventory are not of equal importance in terms of:
– dollars invested
– profit potential
– sales or usage volume
– stock-out penalties
123
ABC Classification System
Classifying inventory according to some measure of importance and allocating control efforts accordingly.
AA - very important
BB - mod. important
CC - least important Annual $ value of items
AA
BB
CC
High
Low
Low HighPercentage of Items
124
Classify inventory into 3 categories typically on the basis of the dollar value to the firm $ volume = Annual demand x Unit cost
A class, B class, C class Policies based on ABC analysis– Develop class A suppliers more carefully – Give tighter physical control of A items– Forecast A items more carefully
ABC Analysis
125
% of Inventory Items
Classifying Items as ABC
0
20
40
60
80
100
0 50 100
% Annual $ Usage
AABB
CC
Class % $ Vol % ItemsA 70-80 5-15B 15 30C 5-10 50-60
126
ABC Classification
11 $ 60$ 60 909022 350350 404033 3030 13013044 8080 606055 3030 10010066 2020 18018077 1010 17017088 320320 505099 510510 6060
1010 2020 120120
PARTPART UNIT COSTUNIT COST ANNUAL USAGEANNUAL USAGE
127
ABC Classification
11 $ 60$ 60 909022 350350 404033 3030 13013044 8080 606055 3030 10010066 2020 18018077 1010 17017088 320320 505099 510510 6060
1010 2020 120120
PARTPART UNIT COSTUNIT COST ANNUAL USAGEANNUAL USAGETOTAL % OF TOTAL % OF TOTALPART VALUE VALUE QUANTITY % CUMMULATIVE
9 $30,600 35.9 6.0 6.08 16,000 18.7 5.0 11.02 14,000 16.4 4.0 15.01 5,400 6.3 9.0 24.04 4,800 5.6 6.0 30.03 3,900 4.6 10.0 40.06 3,600 4.2 18.0 58.05 3,000 3.5 13.0 71.0
10 2,400 2.8 12.0 83.07 1,700 2.0 17.0 100.0
$85,400
128
ABC Classification
11 $ 60$ 60 909022 350350 404033 3030 13013044 8080 606055 3030 10010066 2020 18018077 1010 17017088 320320 505099 510510 6060
1010 2020 120120
PARTPART UNIT COSTUNIT COST ANNUAL USAGEANNUAL USAGETOTAL % OF TOTAL % OF TOTALPART VALUE VALUE QUANTITY % CUMMULATIVE
9 $30,600 35.9 6.0 6.08 16,000 18.7 5.0 11.02 14,000 16.4 4.0 15.01 5,400 6.3 9.0 24.04 4,800 5.6 6.0 30.03 3,900 4.6 10.0 40.06 3,600 4.2 18.0 58.05 3,000 3.5 13.0 71.0
10 2,400 2.8 12.0 83.07 1,700 2.0 17.0 100.0
$85,400
AA
BB
CC
129
ABC Classification
11 $ 60$ 60 909022 350350 404033 3030 13013044 8080 606055 3030 10010066 2020 18018077 1010 17017088 320320 505099 510510 6060
1010 2020 120120
PARTPART UNIT COSTUNIT COST ANNUAL USAGEANNUAL USAGETOTAL % OF TOTAL % OF TOTALPART VALUE VALUE QUANTITY % CUMMULATIVE
9 $30,600 35.9 6.0 6.08 16,000 18.7 5.0 11.02 14,000 16.4 4.0 15.01 5,400 6.3 9.0 24.04 4,800 5.6 6.0 30.03 3,900 4.6 10.0 40.06 3,600 4.2 18.0 58.05 3,000 3.5 13.0 71.0
10 2,400 2.8 12.0 83.07 1,700 2.0 17.0 100.0
$85,400
AA
BB
CC
% OF TOTAL % OF TOTALCLASS ITEMS VALUE QUANTITY
A 9, 8, 2 71.0 15.0B 1, 4, 3 16.5 25.0C 6, 5, 10, 7 12.5 60.0
130
ABC Classification
100 100 –
80 80 –
60 60 –
40 40 –
20 20 –
0 0 –| | | | | |00 2020 4040 6060 8080 100100
% of Quantity% of Quantity
% o
f V
alu
e%
of
Val
ue
AA
BBCC
131
• Inventory accuracy refers to how well the inventory records agree with physical count.
• Cycle counting refers to Physical Count of items in inventory.
• Used often with ABC classification– While A items are counted most often (e.g.,
daily), C items are counted the least frequently.
Inventory Accuracy and Cycle Counting
132
Last Words
Inventories have certain functions.
But too much inventory
- Tends to hide problems
- Costly to maintain
So it is desired
• Reduce lot sizes
• Reduce safety stocks