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Deterministic Inventory Models
(PartII)
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Outline
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Inventory Models
Continuous Review Single Item Models
Basic EOQ model
Economic production quantity model
Quantity discount model
Determining Safety Stock and Reorder Point
For variable demand and constant lead time
For variable demand and variable lead time
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Outline (Contd.)
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Impact of Service Level on Safety Stock
Determining Order Quantity for Periodic Review Models
For variable demand and constant lead time
For variable demand and variable lead time
Visual System
Two-bin system
One-bin system
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Introduction
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The objective of inventory management is to provide the
required level of customer service and to reduce the sum of
all costs involved. To achieve these objectives, two basic
questions must be answered:
How much should be ordered at one time?
When should an ordered be placed?
Management must establish decision rules to answer these
question so that inventory management personnel know
when to order and how much.
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Inventory Models
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Inventory Models
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Inventory control problem is closely related to the demand
pattern. Demand is the principle factor in the design of
inventory model.
i. Deterministic
a) Static demand: constant over time
b) Dynamic demand: Demand is known withcertainty but varies from one time period to
other.
ii. Probabilistic
a) Stationary demand: Demand probability densityfunction remains unchanged over time.
b) Non-stationary demand: Demand probability
density function changes with time.
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Inventory Models (Contd.)
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Other classification of an inventory problem is
i. Single item inventory: Handles only one item or SKU.
ii. Multiple item inventory: When the inventories consists
of many items. Such inventory problems may have
different types of limitations such as finance, coststructure, space etc. As the number of restrictions
increases, the inventory problem becomes more
complicated.
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Continuous Review Single Item Models
Basic EOQ ModelProduction Quantity Model
Quantity Discount Model
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EOQ Model
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Economic Order Quantity(EOQ) - The order size for which
total inventory cost is minimum known as EOQ. Acalculation is made which considers demand rate, ordering
cost, holding cost, lead time, and service level.
The function of the EOQ model is to determine the optimal
order size that minimizes total inventory cost. There are
different variations of EOQ model, depending on the
assumptions made about the inventory system. We will
discuss two basic versions:
i. Basic EOQ model
ii. Production quantity model
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Basic EOQ Model
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The basic EOQ model determines optimal order size that
minimizes the sum of carrying costs and ordering costs. The
assumptions considered are:
i. Demand is known with certainty and is constant over
time
ii. The lead time is also known with certainty and constant
over time
iii. No shortages are allowed
iv. The order quantity is received all at once (i. e.instantaneous replenishment).
v. Decisions for one item can be made independently of
decisions for other items
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Basic EOQ Model (Contd.)
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Inventory Pattern in Basic EOQ model
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Basic EOQ Model (Contd.)
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To determine optimal order size two types of cost are
considered viz. ordering cost and inventory carrying cost.
These two costs react inversely to each other. As order size
increases fewer orders are required, causing the ordering
cost to decline, whereas the average amount of inventory
on hand will increase, resulting in an increase in inventory
carrying cost. Thus in effect, the optimal order quantityrepresents a compromise between these two inversely
related costs.
D = Annual demandC0= ordering cost per order
Ch= Inventory carrying cost (or holding cost) per unit item
Q = Economic order quantity
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Basic EOQ Model (Contd.)
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The figure below shows the inverse relationship between
ordering cost and inventory carrying cost.
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Basic EOQ Model (Contd.)
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Inventory carrying cost increases linearly with order size Q,
while annual ordering cost decreases exponentially with
order-size Q.
The optimal order quantity occurs at the point where the
total cost is minimum, which coincide exactly with the point
where the inventory carrying cost curve intersects the
ordering cost curve. The optimal value of Q can be
determined by differentiating the total cost curve with
respect to Q.
Annual ordering cost = C0 * (D/Q)
Average inventory level = (0 + Q)/2 = Q/2
Annual inventory holding cost or carrying cost = Ch* (Q/2)
Total inventory cost (TC) = C0*(D/Q) + Ch*(Q/2)
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Basic EOQ Model (Contd.)
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To determine optimal order,
= 0
Optimal order size = Qopt
=
Optimal number of order per year = Nopt =
Order cycle time =
Q
)TC(
h
0
C
DC2
OptQ
D
OptNdaysworkingofNumber
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Production Quantity Model
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In production quantity model, the order quantity is received
gradually over time and the inventory level is depleted atthe same time it is being replenished.
This situation is commonly found when the inventory user is
also producer, as in a manufacturing operation where parts
are produced by other section to use in assembly operation.
This situation is also can occur when orders are delivered
continuously over time or when a retailer is also the
producer.
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Production Quantity Model (Contd.)
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In order to determine the average inventory level, we
define the following parameters:
Q = order quantity per order
p = daily rate at which the order is received over time, also
known as the production rate
d = the daily rate at which the inventory is depleted
In this model, we are still assuming that no shortages occur.
Hence demand rate cannot exceed production rate i. e.
d p . If d = p, items are consumed as fast as they are
produced, there is no order size.
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Production Quantity Model (Contd.)
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Inventory pattern of production quantity model
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Production Quantity Model (Contd.)
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D = Annual demand
C0= Setup cost per production run (ordering cost)
Ch= Inventory carrying cost (or holding cost) per unit item
Q = Economic order quantity known as lot size
Time required to finish receiving an order = t/ = Q/p
The amount of inventory that will be depleted or used upduring this time period = t/d = (Q/p)d
As a result, the maximum inventory on hand = Q - (Q/p)d
= Q(1- d/p)
Average inventory level = (Q/2) (1- d/p)
Inventory carrying cost = Ch(Q/2) (1- d/p)
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Production Quantity Model (Contd.)
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To determine optimal order,
= 0
Optimum lot size = Qopt=
Length of production run = (Qopt/p)
Number of production runs = Nopt =
Q
)TC(
)p
d
1(C
DC2
h
0
optQ
D
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Quantity Discount Model
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When items are purchase, suppliers often give a discount in
price over a certain volume or quantity. This is done
because of larger orders reduce the suppliers ordering
costs and cheaper transportation.
Many manufacturing companies receive price discounts for
ordering in high volume from supplier. Retail stores receive
price discounts for ordering merchandise in large
quantities.
A quantity discount is a price discount on an item ifpredetermined numbers of units are ordered.
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Quantity Discount Model (Contd.)
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The basic EOQ model can be used to determine the optimal
order size with quantity discounts. The purchase price of
the item being ordered:
D = Annual demand
C0= ordering cost per order
Ch= Inventory carrying cost (or holding cost) per unit itemQ = Economic order quantity
P = per unit price of item
Total cost (TC) =
The basic assumptions considered are instantaneous
replenishment, and no shortages.
PD2
QC
Q
DC
h0
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Quantity Discount Model (Contd.)
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Total cost curve shows step function behaviour
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Quantity Discount Model (Contd.)
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Quantity Discount Model (Contd.)
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Quantity Discount Model (Contd.)
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Determining Safety Stock and Reorder Point
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Determining Safety Stock and Reorder Point
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Safety stock and reorder point calculation depends on the
following factors:
i. Demand type: Fixed demand or probabilistic demand
ii. Service level: Allowable % of stock out
iii. Lead time: Constant lead time or variable lead time
iv. Frequency of reorder: One prefers replenishment to
take place on a continuous basis. Japanese companies
have managed to get their replenishment from suppliers
three to four times a day in trips involving multiple
deliveries or pickups called milk runs.
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Safety Stock
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Inventory level with safety stock
Safety stock: It is an additional inventory that is kept to take
care of demand and supply uncertainty. Safety stock reduces
chances of stockout situation.
Reorder Point
Safety Stock
Inventory
Time
Lead time
Cycle Stock
Avg. inventory
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Safety Stock (Contd.)
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The average inventory for a period (considering safety stock) is
equal to average of opening inventory and ending inventory.
Opening inventory = order quantity (Q) + safety stock (SS)
Ending inventory = safety stock (SS)
Average inventory = Q/2 + SS
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Reorder Point for Basic EOQ Model
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In basic EOQ model we assumed that demand is constant i. e.
rate of demand is same. We also consider that lead time is
constant i. e. we get order quantity in exactly specified timeperiod.
Reorder point: The level of inventory at which a new order is
placed. Determining the reorder point depends on thedemand during the lead time and the safety stock required. If
demand during the lead time is greater than expected, there
will be a stockout unless sufficient safety stock is available.
Let d = demand rate during lead time
L = lead time
Reorder point (R) = d*L
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Capturing Uncertainty in Demand and Supply
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Uncertainty is captured by one of the following measures:
range, standard deviation, and coefficient of variation.
Demand distribution is captured by two parameters: mean
demand and standard deviation of demand. Standard
deviation of demand is essentially captures uncertainty indemand.
Similarly, supply uncertainty has two parameters, average
lead time and standard deviation of lead time.
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Reorder Point with Variable Demand
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Reorder Point with Variable Demand (Contd.)
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Reorder Point with Variable Demand (Contd.)
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Demand distribution during lead time
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Reorder Point with Variable Demand (Contd.)
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The only time a stockout is possible is during the lead time.
Safety stock is needed to cover only that period in which the
demand during lead time is greater than the average.
If z = 1, it provides service level 84% . It indicates (100 -84)=
16% chance of stockout situations. Similarly, z = 1.65,
provides 95% service level. z = 3, provides 99.83% service
level.
Excel function:
For service factor z = 2, service level = NORMDIST(2, 0,1,1) =
0.977 = 97.7%
For 90% service level, service factor = z = NORMSINV(0.90) =
1.28
d i f i bl d d
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Reorder Point for Variable Demand and
Variable Lead Time
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In practice both demand and lead time are variable. In suchsituation, first we have to determine demand distribution
and lead time distribution in same unit. Second we consider
demand and lead time are independent.
To determine the reorder point we have to determine mean
and standard deviation of demand during lead time, which
is also a random variable. Therefore we have to use
conditional distribution to find out standard deviation of
demand.
R d P i t f V i bl D d d
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Reorder Point for Variable Demand and
Variable Lead Time (Contd.)
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Impact of Service Level on Safety Stock
f i l f k
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Impact of Service Level on Safety Stock
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Safety stock level= service factor (z) * Standard deviation of
demand during lead time
If we improves service factor, amount of safety stock
increases. Service factor determine service level. If service
factor improves from 0 to 1, service level increases by 34%and further improve by 1 unit would improves service level
by 14%. At some point improving service factor service level
improves marginally. Thus relationship between safety stock
level and service factor is non-linear.
f S i l S f S k
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Impact of Service Level on Safety Stock (Contd.)
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For 90% service level, service factor = z = NORMSINV(0.90) =
1.28
I f S i L l S f S k
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Impact of Service Level on Safety Stock (Contd.)
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d f k
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How to Reduce Safety Stock
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Managerial decisions to reduce safety stock
Reduction in demand uncertainty: Minimize the deviation
between actual demand and forecast value by using better
forecasting method or by making contracts with known
customers to generate stable demand.
Reduction in supply uncertainty: work with the suppliers to
reduce lead time, reduce suppliers internal processing
time, and faster mode of transport. Otherwise, select
suppliers based on on-time delivery, flexibility, and quality.
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Determining Order Quantity in Periodic
Review Model
P i di R i S t
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Periodic Review System
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In a continuous review system, the inventory position is
monitored continuously so that an order can be placedwhenever the reorder point is reached.
With a periodic review system, the inventory is checked and
reordering is done only at specified time interval. Forexample, inventory may be checked and orders placed on a
weekly, biweekly, monthly, or some other time interval.
When a firm or business handles multiple products, the
shipping and receiving of orders are easily coordinated
under periodic review system.
P i di R i S t
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Periodic Review System (Contd.)
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Firms running their MRP (material requirement planning)
system place order once in a week, essentially follow theperiodic review system. Similarly, if a retailer places an
order at the time of a salesmansvisit, the periodic review
system is being followed.
While reviewing inventory position, we know that the next
opportunity for ordering will come only after a time interval
(P) and subsequently replenishment will take place after the
lead time (L). In case of continuous review system
vulnerable period is lead time. In case of periodic review
system vulnerable period is review period plus lead time.
Periodic Review System
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Periodic Review System (Contd.)
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Once the inventory in stock is determined after specified time
interval, an order is placed for an amount that will bring back to a
desired inventory level. Hence for periodic review system orderquantity (Q) is variable amount.
Periodic Review: Order Quantity with Variable
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Periodic Review: Order Quantity with Variable
Demand
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Periodic Review: Order Quantity with Variable
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Periodic Review: Order Quantity with Variable
Demand and Variable Lead Time
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Visual System
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Visual System
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Bin 2 Bin 1
When bin 1 is empty,use bin 2 as backup and
place an order
Pick parts
as required
Two-bin system
(Q system)
Visual System
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Visual System (Contd.)
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Single-bin system: When inventory drops to level of Red tag,
place new order
ROP
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