This document discusses inventory systems and models. It begins by defining inventory and the purposes of holding inventory. It then describes different types of inventory including raw materials, work-in-process, and finished goods. The document outlines inventory system models including multi-period models like the fixed-order quantity and fixed-time period models. It also covers single-period models and inventory costs like holding, ordering, and setup costs. The economic order quantity model and how to calculate optimal order quantity is explained. Finally, examples are provided to demonstrate how to apply the economic order quantity and price-break quantity discount models.

1. OBJECTIVES
Inventory System Defined
Types of Inventory
Supply Chain Independent vs. Dependent Demand
Inventory System Models
Management Multi-Period Inventory Models: Basic
Fixed-Order Quantity Models
Inventory Costs
Chapter 5 Multi-Period Inventory Models: Basic
Inventory Control Fixed-Time Period Model
Single-Period Inventory Model
Miscellaneous Systems and Issues
Inventory System Inventory
Inventory is the stock of any item or
resource used in an organization and One of the most expensive assets
can include: raw materials, finished of many companies representing as
products, component parts, supplies, much as 50% of total invested
and work-in-process capital
An inventory system is the set of
policies and controls that monitor levels Inventory managers must balance
of inventory and determines what levels inventory investment and customer
should be maintained, when stock service
should be replenished, and how large
orders should be

2. Purposes of Inventory Types of Inventory
1. To maintain independence of operations Raw material
2. To meet variation in product demand Purchased but not processed
Work-in-process
Work- in-
3. To allow flexibility in production
Undergone some change but not completed
scheduling
A function of cycle time for a product
4. To provide a safeguard for variation in Maintenance/repair/operating (MRO)
raw material delivery time Necessary to keep machinery and processes
5. To take advantage of economic purchase- productive
order size Finished goods
Completed product awaiting shipment
Cycle Inventory
Types of Inventory-2
Inventory that varies directly with lot size.
Lot size varies with elapsed time between
Cycle Inventory orders.
Safety Stock Inventory The quantity ordered must meet the demand
Anticipatory Inventory during the ordering period.
Long gaps in the ordering period will require
Pipeline Inventory larger cycle inventory.
The inventory may vary between order size Q to
zero just before the new lot is delivered.
Average inventory size is therefore Q/2

3. Safety Stock Inventory Anticipation Inventory
Safety stock inventory protects against Inventory used to absorb uneven rate of
uncertainties in demand, lead time, and demand or supply
supply. Predictable seasonal demand pattern may
justify anticipation inventory.
It ensures that operations are not
Uneven demand often makes the firm to
disrupted when problems occur. stockpile during low production demand to
To build safety stock an order is placed make better use of production facilities and
earlier than the item is needed or the avoid varying output rates and labor force.
ordered quantity is larger than the Uncertainties such as threatened strikes,
quantity required till the next delivery problem at suppliers facilities etc also justify
schedule. anticipation inventory.
Pipeline Inventory Independent vs. Dependent Demand
Inventory moving from point to point in the Independent Demand (Demand for the final end-
material flow system is called pipeline product or demand not related to other items)
inventory
- from suppliers to plant, from one Finished
operation to the next in processing, from product
plant to distribution center and from
Dependent
distribution center to retailer Demand
Pipeline Inventory between two points, can be (Derived demand
expressed in terms of lead time and average items for
demand (d) during the lead time (L). E(1) component
parts,
Pipeline Inventory = dL subassemblies,
Component parts raw materials,
etc)

4. Inventory Systems Models Inventory Models for
•• Multi-Period Inventory Models
Multi-Period Inventory Models
-- Fixed-Order Quantity Models
Independent Demand
Fixed-Order Quantity Models
Event triggered (Example: running out of
Event triggered (Example: running out of
stock)
stock) Need to determine when and how
-- Fixed-Time Period Models
Fixed-Time Period Models much to order
Time triggered (Example: Monthly sales
Time triggered (Example: Monthly sales
call by sales representative)
call by sales representative)
Basic economic order quantity
•• Single-Period Inventory Models
Single-Period Inventory Models
-- One time purchasing decision (Example:
One time purchasing decision (Example: Production order quantity
vendor selling t-shirts at a football game)
vendor selling t-shirts at a football game)
-- Seeks to balance the costs of inventory
Quantity discount model
Seeks to balance the costs of inventory
overstock and under stock
overstock and under stock
Holding, Ordering, and Holding Costs
Setup Costs
•Housing costs (including rent or depreciation,
Holding costs - the costs of holding operating costs, taxes, insurance)
or “carrying” inventory over time
•Material handling costs (equipment lease or
Ordering costs - the costs of depreciation, power, operating cost)
placing an order and receiving
goods •Labor cost
Setup costs - cost to prepare a •Investment costs (borrowing costs, taxes, and
machine or process for insurance on inventory)
manufacturing an order
•Pilferage, space, and obsolescence

5. Multi-Period Models: Multi-Period Models:
Fixed-Order Quantity Model Fixed-Order Quantity Model
Assumptions Model Assumptions (Contd.)
Demand for the product is constant Inventory holding cost is based on
and uniform throughout the period average inventory
Lead time (time from ordering to Ordering or setup costs are constant
receipt) is constant
All demands for the product will be
Price per unit of product is constant satisfied (No backorders are allowed)
Basic Fixed-Order Quantity Model and Reorder Cost Minimization Goal
Point Behavior By adding the item, holding, and ordering costs
By adding the item, holding, and ordering costs
together, we determine the total cost curve, which in
together, we determine the total cost curve, which in
1. You receive an order quantity Q. 4. The cycle then repeats. turn is used to find the Qopt inventory order point that
turn is used to find the Qopt inventory order point that
minimizes total costs
minimizes total costs
Number Total Cost
of units C
on hand Q Q Q O
S Holding
T
R Costs
L Annual Cost of
2. You start using L Items (DC)
them up over time. 3. When you reach down to
Time a level of inventory of R, Ordering Costs
R = Reorder point
Q = [Economic] order quantity you place your next Q
QOPT
L = Lead time sized order.
Order Quantity (Q)

6. D
The EOQ Model Annual setup cost =
D
Q
S The EOQ Model Annual setup cost =
Q
S
Q
Annual holding cost = H
2
Q = Number of pieces per order
Q* = Optimal number of pieces per order (EOQ) Q = Number of pieces per order
D = Annual demand in units for the Inventory item Q* = Optimal number of pieces per order (EOQ)
S = Setup or ordering cost for each order D = Annual demand in units for the Inventory item
H = Holding or carrying cost per unit per year S = Setup or ordering cost for each order
H = Holding or carrying cost per unit per year
Annual setup cost = (Number of orders placed per year)
year)
x (Setup or order cost per order)
(Setup order) Annual holding cost = (Average inventory level)
level)
x (Holding cost per unit per year)
(Holding year)
Annual demand Setup or order
= Order quantity
Number of units in each order cost per order = (Holding cost per unit per year)
(Holding year)
2
= D (S)
(S
Q = Q (H)
(H
2
The EOQ Model Basic Fixed-Order Quantity (EOQ) TC=Total annual
TC=Total annual
cost
Model Formula cost
D =Demand
D =Demand
Q = Number of pieces per order Total Annual Annual Annual C =Cost per unit
C =Cost per unit
Q* = Optimal number of pieces per order (EOQ) Q =Order quantity
Annual = Purchase + Ordering + Holding Q =Order quantity
D = Annual demand in units for the Inventory item S =Cost of placing
S = Setup or ordering cost for each order
Cost Cost Cost Cost S =Cost of placing
an order or setup
an order or setup
H = Holding or carrying cost per unit per year
cost
cost
R =Reorder point
R =Reorder point
Optimal order quantity is found when annual setup cost
equals annual holding cost L =Lead time
L =Lead time
H=Annual holding
H=Annual holding
D
D
S =
Q
2
H Annual setup cost =
Q
S
D Q and storage cost
and storage cost
Solving for Q*
Q
Annual holding cost =
Q
H TC = DC + S+ H per unit of inventory
per unit of inventory
2DS = Q2H 2
Q 2
Q2 = 2DS/H
Q* = 2DS/H

7. Deriving the EOQ The Economic Ordering Quantity (EOQ)
2DS
2DS = 2(Annual Demand)(Order or Setup Cost)
2(Annual Demand)(Order or Setup Cost)
QOPT =
QOPT = H =
H Annual Holding Cost
Annual Holding Cost
__
We also need a
We also need a R eorder point, R = d L
R eorder point, R = d L
reorder point to
reorder point to _
tell us when to
tell us when to d = average daily demand (constant)
place an order
place an order
L = Lead time (constant)
EOQ Example-1 EOQ Example-1a
Determine expected number of orders if:
Determine optimal number of units to order
D = 1,000 units D = 1,000 units Q* = 200 units
S = $10 per order S = $10 per order
H = $.50 per unit per year H = $.50 per unit per year
2DS Expected
Q* = Demand D
H number of = N = Order quantity = Q*
*
orders
2(1,000)(10) N=
1,000
= 5 orders per year
Q* = = 40,000 = 200 units 200
0.50

8. EOQ Example- 1b EOQ Example- 1c
Determine time between orders if: Determine carrying cost if:
D = 1,000 units Q*= 200 units
Q*= D = 1,000 units Q* = 200 units
S = $10 per order N= 5 orders per year
N= S = $10 per order N = 5 orders per year
H = $.50 per unit/yr working days= 250 days/yr H = $.50 per unit per year T = 50 days
Number of working Total carrying cost = Setup cost + Holding cost
Expected days per year D Q
time between = T = TCC = S + H
orders N Q 2
250 1,000 200
TCC = ($10) + ($.50)
T= = 50 days between orders 200 2
5
TCC = (5)($10) + (100)($.50) = $50 + $50 = $100
Holding cost is often given as a
fraction of unit cost
Holding cost as a fraction of unit cost
Quantity Discount Model or 2DS 2(Annual Demand)(Or der or Setup Cost)
Q OPT = =
Price-Break Model iC Annual Holding Cost
i = percentage of unit cost attributed to carrying inventory
C = cost per unit

9. Price-Break Example- 2
Price-Break Model Formula Problem Data (Part 1)
or Quantity Discount Model A company has a chance to reduce their inventory
A company has a chance to reduce their inventory
ordering costs by placing larger quantity orders using
ordering costs by placing larger quantity orders using
Based on the same assumptions as the EOQ model, the
price-break model has a similar Qopt formula: the price-break order quantity schedule below. What
the price-break order quantity schedule below. What
should their optimal order quantity be if this company
should their optimal order quantity be if this company
purchases this single inventory item with an e-mail
purchases this single inventory item with an e-mail
2DS 2(Annual Demand)(Or der or Setup Cost) ordering cost of $4, a carrying cost rate of 2% of the
ordering cost of $4, a carrying cost rate of 2% of the
Q OPT = =
iC Annual Holding Cost inventory cost of the item, and an annual demand of
inventory cost of the item, and an annual demand of
10,000 units?
10,000 units?
i = percentage of unit cost attributed to carrying inventory
C = cost per unit Order Quantity units) Price/unit($)
0 to 2,499 $1.20
Since “C” changes for each price-break, the formula above 2,500 to 3,999 $1.00
will have to be used with each price-break cost value 4,000 or more $0.98
Price-Break Example-2 Solution (Part 2) Price-Break Example -3 Solution (Part 3)
First, plug data into formula for each price-break value of “C” Since the feasible solution occurred in the first price-
Since the feasible solution occurred in the first price-
break, it means that all the other true Qopt values occur
break, it means that all the other true Qopt values occur
Annual Demand (D)= 10,000 units Carrying cost % of total cost (i)= 2%
Cost to place an order (S)= $4 Cost per unit (C) = $1.20, $1.00, $0.98
at the beginnings of each price-break interval. Why?
at the beginnings of each price-break interval. Why?
Next, determine if the computed Qopt values are feasible or not Because the total annual cost function is
Because the total annual cost function is
Total a “u” shaped function
annual a “u” shaped function
Interval from 0 to 2499, the 2DS 2(10,000)( 4)
Qopt value is feasible Q OPT = = = 1,826 units costs
iC 0.02(1.20) So the candidates
So the candidates
for the price-
for the price-
Interval from 2500-3999, the 2DS 2(10,000)( 4)
Qopt value is not feasible Q OPT = = = 2,000 units breaks are 1826,
breaks are 1826,
iC 0.02(1.00) 2500, and 4000
2500, and 4000
Interval from 4000 & more, 2DS 2(10,000)( 4) units
units
the Qopt value is not feasible Q OPT = = = 2,020 units
iC 0.02(0.98)
0 1826 2500 4000 Order Quantity

10. Price-Break Example -3 Solution (Part 4)
Next, we plug the true Qopt values into the total cost
Next, we plug the true Qopt values into the total cost
annual cost function to determine the total cost under Price-Break Example -3 Solution (Part 5)
annual cost function to determine the total cost under
each price-break
each price-break
D Q
TC = DC + S + iC
Q 2
TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20)
TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20)
= $12,043.82
= $12,043.82
TC(2500-3999)= $10,041
TC(2500-3999)= $10,041
TC(4000&more)= $9,949.20
$9849,20
TC(4000&more)= $9,949.20
Finally, we select the least costly Qopt,,which is this
Finally, we select the least costly Qopt which is this
problem occurs in the 4000 & more interval. In
problem occurs in the 4000 & more interval. In
summary, our optimal order quantity is 4000 units
summary, our optimal order quantity is 4000 units
Production Order Quantity
Model
In EOQ Model, We assumed that the
entire order was received at one
Production Order Quantity Model time.
However, some business firms may
receive their orders over a period of
time.

11. Production Order Quantity Model
Production Order Quantity Model
Such cases require a different inventory
model. In these cases inventory is being This version of the EOQ model is known as
used while new inventory is still being “Noninstantaneous Receipt Model” also
received and the inventory does not build referred to as the “Gradual Usage Model ” and
up immediately to its maximum point. “Production Order Quantity Model”. In this,
noninstantaneous receipt model, the order
Instead, it builds up gradually when
quantity is received gradually over time, and the
inventory is received faster than it is
inventory level is depleted at the same time it is
being used; then it declines to its lowest being replenished.
level as incoming shipments stop and the
use of inventory continues. Here, we take into account the daily production
rate and daily demand/input rate.
Production Order Quantity Model Production Order Quantity Model
Inventory
Maximum Inventory
Production
occurs at a Demand
rate of p occurs at a
rate of d
time
t
t

12. Production Order Quantity Production Order Quantity
Model Model
Since this model is especially suitable for p: Daily Production rate (units / day)
production environments, It is called d: Daily demand rate (units / day)
Production Order Quantity Model. t: Length of the cycle in days.
Here, we use the same approach as we H: Annual holding cost per unit
used in EOQ model.
Lets define the following:
Production Order Quantity Model Production Order Quantity
Model
Average Holding Cost = (Average In the period of production (until the end of
Inventory) . H each t period):
Max. Inventory = (Total Produced) – (Total Used)
= (Max. Inventory / 2) . H = p.t - d.t
Here, Q is the total units that are
produced. Therefore,
Q = p.t t = Q/p

13. Production Order Quantity Production Order Quantity
Model Model
If we replace the values of t in the Max. Ann. Holding Cost =(Max. Inventory / 2) . H
Inventory formula: Annual Holding Cost = Q/2 (1 – d/p) . H
Max. Inventory = p (Q/p) - d (Q/p)
= Q - dQ/p = Q (1 – d/p) Annual Setup Cost = (D/Q) . S
Production Order Quantity Production Order Quantity
Model Model
Now we will set
Annual Holding Cost = Annual Setup Cost
Q/2 (1 – d/p) . H = (D/Q) . S

14. Production Order Quantity Production Order Quantity-
Model Example-5
D = 1,000 units p = 8 units per day
This formula gives us the optimum S = $10 d = 4 units per day
production quantity for the Production H = $0.50 per unit per year
Order Quantity Model.
2DS
Q* =
H[1 - (d/p)]
It is used when inventory is consumed as
it is produced. Q* =
2(1,000)(10)
= 80,000
0.50[1 - (4/8)]
= 282.8 or 283 units
How Important is the Item?
Segmentation of Inventory
- Not all inventory is created equally
- Different classes of inventory
Miscellaneous Systems and - Result in different levels of profitability /revenue
- Have different demand patterns and magnitudes
Issues
- Require different control policies
ABC Analysis
Commonly used in practice
Classify items by revenue or value
Combination of usage, sales price, etc.

15. ABC Analysis ABC Analysis
Identify the items that management
should spend time on
Prioritize items by their value to firm
Create logical groupings
Adjust as needed
ABC Analysis Miscellaneous Systems:
What is different between the classes? Bin Systems
A Items
Very few high impact items are included Two-Bin System
Require the most managerial attention and review
Expect many exceptions to be made
B Items Order One Bin of
Many moderate impact items (sometimes most) Inventory
Automated control w/ management by exception
Rules can be used for A (but usually too many exceptions) Full Empty
C Items One-Bin System
Many if not most of the items that make up minor impact
Control systems should be as simple as possible
Reduce wasted management time and attention Order Enough to
Group into common regions, suppliers, end users
Refill Bin
But these are arbitrary classifications
Periodic Check

16. Miscellaneous Systems: Inventory Accuracy and Cycle Counting
OptionalInventory Level, M System
Maximum Replenishment
Inventory accuracy refers to how
q=M-I
well the inventory records agree
Actual Inventory Level, I with physical count
M
Cycle Counting is a physical
I inventory-taking technique in which
inventory is counted on a frequent
Q = minimum acceptable order quantity basis rather than once or twice a
year
If q > Q, order q, otherwise do not order any.
Question
On average, I sell 150,000 units a year, which I
obtain from a wholesaler. I estimate that the
cost to me of placing an order is $50 with the Supply Chain
average inventory storage cost being 20% per
year of the cost of a unit ($5). Management
1. What would be the optimal order quantity?
2. I currently order 5 times a year. How much Inventory Control Part 2
would I save by switching to the optimal order
Safety Stock, Fixed Period Model
quantity as compared with my current policy of
ordering 5 times a year? and Single Period Model

17. Planned Shortages with Back-Orders Uncertain Demand
Shortage: when customer demand cannot
be met
Planned shortages could be beneficial
Cost of keeping item is more expensive than
the profit from selling it e.g. car
Uncertain Demand- Safety Stock Service Level
A target for the proportion of demand that
Buffer added to on hand is met directly from stock
inventory during lead time
Extra reserved stock
The maximum acceptable probability that
To prevent stock-out a demand can be met from the stock
under uncertain demand For example 90% service level
Safety stock will not 90% chance of meeting demand during lead time or
normally be used, but it is 10% chance of not meeting demand (having back-
available under uncertain order or lost sales)
demand
How much safety stock should we
hold? Judgment on service level

18. Probabilistic Models Probabilistic Models
So far we assumed that demand is One method of reducing stock outs is to
constant and uniform. hold extra inventory (called Safety Stock).
However, In Probabilistic models, demand In this case, we change the ROP formula
is specified as a probability distribution. to include that safety stock (ss).
Uncertain demand raises the possibility of
a stock out (or shortage).
Reorder Level (ROL) = LT x D Safety Stock Example
Reorder Level (ROL) = (LT x D) + Safety Stock
ROP = 50 units Stock-out cost = $40 per unit
Orders per year = 6 Carrying cost = $5 per unit per year
Reorder Number of Units Probability
Level 30 0.2
40 0.2
ROP 50 0.3
60 0.2
70 0.1
1.0
Safety Stock

19. Safety Stock Example Example
ROP = 50 units Stock-out cost = $40 per unit
Orders per year = 6 Carrying cost = $5 per unit per year
A safety stock of 20 units gives the lowest total cost
ROP = 50 + 20 = 70 units
Probabilistic Demand

20. Reorder Point for a Service Level
Using the
Standard
Normal
Probability
Table
Using the Standard Normal Probability Table
=

21. Probabilistic Demand Example- 3: Safety Stock
Demand is variable and lead time is constant
Safety stock, SS: Daily usage at a drug store follows a
= Z × standard deviation of lead time normal distribution with a mean of 500 gm
= Z × σ × √LT and a standard deviation of 50 gm. If the
= Z × σdlt lead time for procurement is 7 days and
Reorder level: the drug store wants a risk of only 2%
ROL = lead time demand + safety stock determine
= LT × D + Z × σ × √LT
where σ = standard deviation of demand per day and a) reorder point and b) safety stock
σdlt = σ × √LT Standard deviation of demand during necessary
lead time
Example-3: Safety Stock Example: Safety Stock using Z-Score
Mean daily demand, D =500 gm/day
Lead Time, LT = 7days Mean Demand in lead period, µL =3500 gm
Standard deviation, σ = 50 gm/day Standard deviation, σ = 50 gm/day
Service level required = 98% or 0.98 σL = σ √ Lt= 50 √7 gm
From normal distribution level Z is determined as z =2.05 Z= 2.05 from Table
ROL = (LT × D)+ z σ √LT X − µL
Z=
= (500 x 7)+ 2.05 * 50 * √7 σL
= 3771 gm where X is a normal random variable
Safety Stock = z σ √LT X=3771gm
= 2.05 * 50 * √7 = 271 gm Safety stock = 3771 gm- 3500 gm =271 gm

22. Periodic Review System
Maximum Inventory Level, M
Supply Chain q=M-I
Management M
Actual Inventory Level, I
I
Inventory Control
Periodic Review System
P-System: Periodic Review System P-System: Periodic Review System-2
In this system, costs are not explicitly In this system, we are interested in actual
considered and order quantity is not fixed. and average consumption over a period of
time i.e. time between two reviews and lead
Time is taken into account and given more time. Order quantity can be computed as
emphasis follows:
Inventory is periodically reviewed at fixed
intervals and any difference between the If L< R then Q= M - I If L> R then Q= M – I - Q ord
present and the last review is made up by
replenishment order.
Where
The order quantity is thus equal to L= Lead Time R = Review Period
replenishment level minus actual inventory M= Replenishment Level in Units I = Inventory on hand in Units
on hand. Q =Quantity to be Ordered Qord= Quantity on order (in pipeline)

23. Example: Fixed Period Inventory Control
System (P-System)
Example-Solution: P-System
L<R
The average Replen. Lvl. = M Replenishment Level, M =
60
monthly Safety Stock (B)+ consumption, D* (Review Time+ Lead Time)
consumption of an M= 20+ 40(1+0.5) = 80 Units
item is 40 units, 40
Inventory on Hand, I = B + consumption/2
Safety Stock is 20
units, review time I = 20+ 40/2 = 40 units
Safety Stock=B
20
is 1 month and R The Order Quantity, Q = M – I
LT
lead time is 15 Q= 80- 40 = 40 Units
days, calculate 1 2 3
replenishment
level M
Example 2: (P-System)
Fixed Order Vs. Periodic Review
Consider a case where Lead time > review time
Buffer/safety stock= 50 units D= 100 units/month Fixed-order quantity Fixed-time period
Review Time= 1 month L= 2 months models–when holding models—when holding
costs are high (usually costs are low (i.e.,
M= replinsh. Lvl. = B +D (1+2) = 50+ 100*3
expensive items or high associated with low-cost
M= 350 Units deprecation rates), or items, low-cost storage),
I = B+D/2 = 50+50 = 100 units when items are ordered or when several items are
from different sources. ordered from the same
Order Qty Q= M – I = 250 units source (saves on order
If Qty already on order is 100 units (review after 1 mth) placement and delivery
charges).
Q= M-I- Qord= 150 units

24. Fixed Order Vs. Periodic Review
A fixed-order quantity The main disadvantage of
system can operate with a fixed-time period
a perpetual count inventory system is that
(keeping a running log of inventory levels must be
every time a unit is
withdrawn or replaced) or
higher to offer the same
protection against
Single-Period Inventory
through a simple two-bin
or flag arrangement
stockout as a fixed-order
quantity system.
Model
wherein a reorder is It also requires a periodic
placed when the safety count and closer
stock is reached surveillance than a fixed-
order quantity system.
Decision under uncertainity & risk Single-Period Inventory Model
In inventory control, sometimes management has to take
In inventory control, sometimes management has to take This model states that we
This model states that we IG is the profit per item
risk under uncertainity, though wanting to keep the risk should stock up to the point IG is the profit per item
risk under uncertainity, though wanting to keep the risk should stock up to the point times the probability of
times the probability of
factor to a minimum.
factor to a minimum. where incremental gain (IG)
where incremental gain (IG) selling ‘x’ items
selling ‘x’ items
is equal to incremental loss
is equal to incremental loss
•• How many World Cup shirts to produce, when the shirts (IL)
(IL) IG= m. P(x)
IG= m. P(x)
How many World Cup shirts to produce, when the shirts
will be of little or no value after the Cup.
will be of little or no value after the Cup.
IL is the cost per item times
IL is the cost per item times m= margin of profit item
m= margin of profit item
•• How many suits to stock for Eid or Xmas season, profit
How many suits to stock for Eid or Xmas season, profit the probability that ‘x’ items P (x)= probability of selling the item
margin is high but the leftover stock will probably be of the probability that ‘x’ items P (x)= probability of selling the item
margin is high but the leftover stock will probably be of will not be sold
will not be sold
C= Cost of the item
C= Cost of the item
no value
no value
IL= C. [1-P(x)].
IL= C. [1-P(x)].
Equating IG& IL and
Equating IG& IL and P(x) =
P(x) = C
C
Single-period inventory model Applies in these cases solving the equation we get:
solving the equation we get: m+C
m+C

25. Single-Period Inventory Model
Single Period Model Example-4 This model states that we
This model states that we
should continue to increase
should continue to increase
Our college basketball team is playing in a the size of the inventory so
the size of the inventory so
tournament game this weekend. Based on our past long as the probability of
long as the probability of Cu
experience we sell on average 2,400 shirts with a
standard deviation of 350. We make $10 on every
selling the last unit added is
selling the last unit added is P≤
shirt we sell at the game, but lose $5 on every shirt equal to or greater than the
equal to or greater than the Co + Cu
not sold. How many shirts should we make for the ratio of: Cu/Co+Cu
ratio of: Cu/Co+Cu
game?
Cu = $10 and Co = $5; P ≤ $10 / ($10 + $5) = .667 Where :
Z.667 = .432 Co = Cost per unit of demand over estimated
therefore we need 2,400 + .432(350) = 2,551 shirts
Cu = Cost per unit of demand under estimated
P = Probability that the unit will be sold
Example-6
Example-5 (Solution) Ahmed Juices makes a variety of juices for on-the-
counter sales. Ahmed uses ice, which he grates in
Where : making these drinks. Ice is supplied to Ahmed in large
Cu blocks, each costing Rs 10. Ice blocks not used during
P≤ Co = Cost per unit of demand over estimated
a day gets wasted as the ice melts and cannot be used
Co + Cu Cu = Cost per unit of demand under estimated the next day. If Ahmed is short of ice blocks on any day,
P = Probabilit y that the unit will be sold he buys them from elsewhere, but at a premium of Rs
5 per block. Each block of ice can be used for 20
Co = Rs 1.5 [(Cost) Loss if demand is overestimated] glasses of juice. The probability distribution for the
demand of ice blocks is as follows
Cu = Rs 2.5 [(Cost) Profit Loss if demand is underestimated] What is the least cost stocking policy for Ahmed
Juices?
Probability of meeting demand is
P ≤ [2.5/(2.5+1.5)] 0.65 at 700 buns. The baker
x ice blocks: 20 21 22 23 24 25 26 27 28
P ≤ 0.625 should make 700 buns. p Probability 0 0.05 0.10 0.20 0.25 0.20 0.15 0.05 0

26. Practice Numerical
The end
Example (Contd.)
Example The present cycle and pipeline inventories are:
The present cycle and pipeline inventories are:
A plant makes monthly shipments of electric Cycle Inventory = Q/2 = 280/2 = 140 drills
Cycle Inventory = Q/2 = 280/2 = 140 drills
A plant makes monthly shipments of electric
drills to a wholesaler in average lot sizes of
drills to a wholesaler in average lot sizes of Pipeline Inventory, dL= (70 drills/week)* (3 weeks) = 210 drills
Pipeline Inventory, dL= (70 drills/week)* (3 weeks) = 210 drills
280 drills. The wholesaler’s average demand
280 drills. The wholesaler’s average demand
is 70 drills a week and the lead time from the
is 70 drills a week and the lead time from the
plant is 3 weeks. The wholesaler must pay for
plant is 3 weeks. The wholesaler must pay for Under the new offer, cycle and pipeline inventories are:
Under the new offer, cycle and pipeline inventories are:
the order the moment it leaves the plant.
the order the moment it leaves the plant. Cycle Inventory = Q/2 = 350/2 = 175 drills
Cycle Inventory = Q/2 = 350/2 = 175 drills
If the wholesaler is willing to increase its
If the wholesaler is willing to increase its Pipeline Inventory, dL= (70 drills/week)* (2 weeks)
Pipeline Inventory, dL= (70 drills/week)* (2 weeks)
purchase quantity to 350 units, the plant will
purchase quantity to 350 units, the plant will = 140 drills
= 140 drills
guarantee a lead time of two weeks. What is
guarantee a lead time of two weeks. What is
effect on cycle and pipeline inventories? Under the new offer, cycle inventory increases by 25% but
Under the new offer, cycle inventory increases by 25% but
effect on cycle and pipeline inventories? pipeline inventories reduce by 33% (Decision Point)
pipeline inventories reduce by 33% (Decision Point)

27. Benefit of Better Inventory Control Example -1
A firm's inventory turnover (IT) is 4 times on a Fleming sells distributor rebuild kits used on
cost of goods sold (COGS) of $800,000. Ford V-8 engines. Fleming purchases these kits
Through better inventory control, inventory for $20 and sells about 250 kits a year. Each
time Fleming places an order, it costs him $25 to
turnover is improved to 8 times while the cover paperwork. He estimated that the cost of
COGS remains the same, a substantial holding a rebuild kit in inventory is about $3.5
amount of funds is released from inventory. per kit per year.
What is the amount released?
a) What is the economic order quantity
b) How many times per year will Fleming place
$ 100,000 is released an order?
Example -1 (Contd.) EOQ Example (2) Problem Data
S = Cost of placing order = $ 25
D= Annual demand = 250 units/year Given the information below, what are the EOQ and
Given the information below, what are the EOQ and
H= Annual per-unit carrying cost =$3.5 per reorder point?
reorder point?
kit/year
Annual Demand = 1,000 units
Q = order quantity
Days per year considered in average
Qopt= √ [2 S D/H] daily demand = 365
Cost to place an order = $10
Qopt= √ [(2*25*250)/3.5] Holding cost per unit per year = $2.50
= 59.75 round to 60 kits Lead time = 7 days
Orders per year =D/Qopt = 250/59.75 Cost per unit = $15
= 4.18

28. EOQ Example (2) Solution EOQ Example (3) Problem Data
2DS 2 (1,00 0 )(1 0) Determine the economic order quantity
Q O PT = = = 8 9.443 u nits o r 90 un its Determine the economic order quantity
H 2.50 and the reorder point given the following…
and the reorder point given the following…
1,000 units / year
d = = 2.74 units / day
365 days / year Annual Demand = 10,000 units
Days per year considered in average daily
_
R e order po int, R = d L = 2.7 4units / d ay (7d ays) = 1 9.18 or 20 u n its
demand = 365
Cost to place an order = $10
Holding cost per unit per year = 10% of cost
In summary, you place an optimal order of 90 units. In
In summary, you place an optimal order of 90 units. In
the course of using the units to meet demand, when
per unit
the course of using the units to meet demand, when
you only have 20 units left, place the next order of 90
you only have 20 units left, place the next order of 90
Lead time = 10 days
units.
units. Cost per unit = $15
EOQ Example (3) Solution Example- 8
2D S 2(10,000 )(10) Demand for Deskpro computer at Best Buy is 1000 units
Q OPT = = = 365.148 units, or 366 units per month. Best Buy incurs a fixed order placement,
H 1.50
transportation and receiving cost of $4000 each time an
order is placed. Each computer costs Best Buy $500 and
10,000 units / year the retailer has a holding cost of 20%. Evaluate the
d= = 27.397 units / day
365 days / year number of computers that the store manager should
order in each replenishment lot.
_
R = d L = 2 7 .3 9 7 u n its / d ay (10 d ays) = 2 7 3 .9 7 o r 2 7 4 u n its
Annual Demand, D = 1000 x12 = 12000 units
Order cost per lot, S = $4000
Place an order for 366 units. When in the course of
Place an order for 366 units. When in the course of
using the inventory you are left with only 274 units, Unit Cost per computer, C =$500
using the inventory you are left with only 274 units,
place the next order of 366 units. Holding cost per year as a fraction of the inv. Value, h = 0.2
place the next order of 366 units.

29. Example-8 Solved Example-9
Q opt = √ 2 * 12000 * 4000 In the above example, the manager at Best Buy
0.2 * 500 would like to reduce the lot size from 980 to 200.
For this lot size to be optimal, the store manager
= 980 units wants to evaluate how much the order cost per
Other Info lot should be reduced.
Cycle Inventory = Qopt/2 = 980/2 = 490 Desired Qopt = 200 units
No. of orders/year = D/Q = 12000/980 = 12.24 Annual Demand, D = 1000 x12 = 12000 units
Annual ordering & holding costs =(D/Q)*S + (Q/2)hC New Order cost per lot, S = ?
= $97,980 Unit Cost per computer, C =$500
Average Flow time= Q/2D = 490/12000 = 0.041year Holding cost per year as a fraction of the inv. Value, h = 0.2
= 0.49 months
Example-9 (Contd.) Problem-10
S = H [Qopt]2/2D The Acer Co. sells 10,000 units per year.
H =hC= 0.2*500 The cost of placing one order is $50 and it
S = [0.2*500* 2002]/ [2*12000] costs $4 per year to carry one unit of
inventory. What is Acer’s EOQ?
S = $166.7
THUS THE STORE MANAGER AT BEST BUY WOULD HAVE TO
REDUCE THE ORDER COST PER LOT FROM $4000 TO $166.7 FOR
A LOT SIZE OF 200 TO BE OPTIMAL