L03-05: Multi Period Inventory Systems SCM 2011

Introduction to a few Terms

• Customer – anyone or anything that creates demand for items

• Supplier – anyone or anything that replenishes or adds to stock

Inventory Management • Stock – all the goods and materials that are stored by an organization
until they are needed
• Inventory – is a list of the items held in stock (often taken as being the
stock itself)
Multi-Period Inventory Systems
• Consumables – stocks of materials needed to support operations, but
which do not form part of the final product, such as oil, paper, cleaners,
etc.
• Item – a single article that is kept in stock – it is one entry in the
Sushil Kumar, PhD inventory
IIM Lucknow • Stock Keeping Unit (SKU) – an alternative name for item
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Introduction to a few Terms Introduction to a few Terms

• Spare parts – items held in stock as replacements to keep machinery, • Safety stock – a reserve of materials that is not normally needed, but is
equipment, etc working properly held to cover unexpected circumstances
• Item coding – an arrangement for giving every package of material • Service level – a measure of the proportion of customer demand met from
moved an identifying tag, usually a bar code or magnetic strip stock (or some equivalent measure) (Fill Rate)
• Purchasing – part of procurement responsible for actually buying
• Unit – the standard size or quantity of a stock item
materials (and often used to mean the same as procurement)
• Unit cost – cost of buying (or acquiring) each unit of an item • Procurement –the function responsible for acquiring all the materials
• Unitization – putting materials into standard packages (typically on needed by an organization
pallets or in containers) to ease movement • Procurement cycle – sequence of activities needed to acquire materials
• Vendor-managed inventory – has suppliers managing both their own • Replenishment – putting materials into stock to replace units that have
stocks and those held further down the supply chain been used
• Stocktaking – periodic checks to find differences between recorded and
actual stock levels

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Introduction to a few Terms Introduction to a few Terms

• Newsboy problem – a standard problem of finding the best order size for
• Destock – reduce the amount of stock held a single stock cycle, with uncertain demand
• Cycle-counting – where stock is checked at regular intervals, with a • Electronic data interchange (EDI) – a method of transferring data
small proportion of items typically checked every week directly between remote computers
• Dependent demand methods – assume the demand for an item is • Electronic fund transfer (EFT) – a method of automatically debiting a
directly related to the demand for other items, with this relationship used customer’s bank account and crediting the money to a supplier’s account
to control stocks
• Electronic point of sales (EPOS) – a system that records transactions at
• Lost sales – when customer demand cannot be met, and the customer
withdraws their demand a cash register and transfers the information to stock control and other
• MRP – material requirements planning functions
• Order – a message from an organization to a supplier requesting a
• MRP II – manufacturing resource planning
delivery of materials
• Lot sizing – combining several small orders into larger ones

http://www.inventoryops.com/dictionary.htm
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What is Inventory? Inventory: A Necessary Evil

• Inventory is the stock of any resource (man, machine,
• Inventory is the stock of any item or resource used material) of value, kept for future use.
in an organization and can include: raw materials, • Inventory is must to absorb the shock of demand forecast
finished products, component parts, supplies, and error, to permit more effective use of facility and staff, and
work-in-process isolate one part of system from other.
Input Finished Demand
Prodn. Goods +
+ Inventory Uncertainty
Uncertainty system Inventory

O
In-process U
In- process T
Input Inventory P
Inventory U
T

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Examples of inventories: Purposes of Inventory

• Manufacturing firms carry supplies of raw materials, 1. To maintain independence of operations

purchased parts, finished items, spare parts, tools,.... 2. To meet variation in product demand
• Department stores carry clothing, furniture, stationery,
3. To allow flexibility in production scheduling
appliances,...
• Hospitals stock drugs, surgical supplies, life-monitoring 4. To provide a safeguard for variation in raw material delivery
equipment, sheets, pillow cases,... time
• Supermarkets stock fresh and canned foods, packaged and 5. To take advantage of economic purchase-order size
frozen foods, household supplies,..
6. To help hedge against price increases

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Types of Inventory
Types of Inventories
• Seasonal Inventory: Seasonality in demand is absorbed
using inventory
Manufacturing facility • Decoupling Inventory: Complexity of production control is
reduced by splitting manufacturing into stages and
Supplier Step 1 Step 2 Customer maintaining inventory between these stages
`
• Cyclic Inventory: Periodic replenishment causes cyclic
inventory
In-Transit Raw Finished • Pipeline Inventory: Exists due to lead time
WIP
Material Goods • Safety Stock: Used to absorb fluctuations in demand due to
` ` ` ` uncertainty

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Cyclic, Pipeline and Safety Stocks
A graphical illustration Types of Inventories

• Raw materials & purchased parts

Cyclic • Partially completed goods called work in progress
Stock
Quantity

• Finished-goods inventories (manufacturing firms)

Pipeline inventory
or merchandise (retail stores)
Safety stock
L

• Replacement parts, tools, & supplies

Time

Cyclic inventory, pipeline inventory and safety stocks are critically • M R O ( Maintenance/Repairs/Operating supplies)
linked to “how much” and “when” decisions in inventory planning
• Goods-in-transit to warehouses or customers
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Inventory System Defined Functions of Inventory

• An inventory system is the set of policies and controls that
monitor levels of inventory and determines what levels should • To meet anticipated demand
be maintained, when stock should be replenished, and how
• To smooth production requirements
large orders should be
• To decouple components of the production-distribution
Inventory Management system
• To protect against stockouts
Good inventory management is essential for successful • To take advantage of order cycles
operation in an organization
• To hedge against price increases or to take advantage of
Reasons: quantity discounts
• The amount of money inventory represents • To permit operations
• The impact that inventories have on the daily operations
of an organization.
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Inventory Management:
Objectives of Inventory Control
Overall Objective
• Inadequate control of inventories can result in both
under- and overstocking of items... Achieve satisfactory levels of customer
service while keeping inventory costs
• Understocking results in missed deliveries, lost sales, within reasonable bounds.
dissatisfied customers, and production delays.
• Overstocking ties up funds that might be more This means, the manager tries to achieve a balance in stocking.
productive elsewhere.
Two fundamental decisions are to be made:
• Timing of the order

• Size of the order.

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Requirements for Effective Inventory Management

(Systems) Inventory Costs

• A system to keep track of inventory

• Holding (or carrying) costs
• A reliable forecast of demand • Costs for storage, handling, insurance, etc

• Knowledge of lead times • Setup (or production change) costs

• Costs for arranging specific equipment setups, etc
• Reasonable estimates of
• Ordering costs
• Holding costs • Costs of someone placing an order, etc
• Ordering costs • Shortage costs
• Costs of canceling an order, etc
• Shortage costs

• A classification system

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Costs in Inventory Planning Computation of Carrying Cost
Carrying Cost
An Illustration
• Interest for short-term borrowals for working capital
• Cost of stores and warehousing
• Administrative costs related to maintaining and accounting
for inventory
• Insurance costs, cost of obsolescence, pilferage, damages
and wastage
• All these costs are directly related to the level of inventory

*The percentage for obsolescence and damages and so on are estimates based on
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Costs in Inventory Planning Computation of Ordering Cost

Ordering Cost

• Search and identification of appropriate sources of supply An Illustration

• Price negotiation, contracting and purchase order generation
• Follow-up and receipt of material
• Eventual stocking in the stores after necessary accounting
and verification
• A larger order quantity will require less number of orders to
meet a known demand and vice versa

Cost of carrying and cost of ordering are fundamentally two

opposing cost structures in inventory planning

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Costs in Inventory Planning Independent vs. Dependent Demand
Shortage Cost
Inventory: a stock or store of goods
• Costs arising out of pushing the order back and Independent Demand
rescheduling the production system to accommodate these
changes
A Dependent Demand
• Rush purchases, uneven utilisation of available resources Dependent Demand
and lower capacity utilisation (Derived demand
items for component
B(4) C(2)
• Missed delivery schedules leading to customer parts,
subassemblies,
dissatisfaction and loss of good will raw materials, etc)
• The effects of shortage are vastly intangible, it is indeed D(2) E(1) D(3) F(2)

difficult to accurately estimate

Independent demand is uncertain.

Dependent demand is certain.
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Inventory Systems Inventory: A Necessary Evil

• Single-Period Inventory Model
• Inventory is the stock of any resource (man, machine,
• One time purchasing decision
material) of value, kept for future use.
(Example: vendor selling T-shirts at a football game)
• Inventory is must to absorb the shock of demand forecast
• Seeks to balance the costs of inventory overstock and under stock
error, to permit more effective use of facility and staff, and
• Multi-Period Inventory Models
isolate one part of system from other.
• Fixed-Order Quantity Models
Input Finished Demand
• Event triggered (Example: running out of stock) Prodn.
+ Inventory Goods +
system Inventory Uncertainty
• Fixed-Time Period Models Uncertainty

• Time triggered (Example: Monthly sales call by sales

O
representative) In-process U
In- process T
Input Inventory P
Inventory U
T

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What factors influence
Three Levels of Inventory Decisions inventory replenishment models?
• Supply Chain Decisions (strategic)
• What are the potential alternatives to inventory?
• How should the product be designed?
• Deployment Decisions (strategic)
• What items should be carried as inventory?
• In what form should they be maintained?
• How much of each should be held and where?
• Replenishment Decisions (tactical/operational)
• How often should inventory status be determined?
• When should a replenishment decision be made?
• How large should the replenishment be?

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Multi-Period Inventory Systems Multi-Period Inventory Systems

These systems ensure the material availability on ongoing basis • Fixed-order quantity models are Event Triggered when
throughout the year.
inventory drops to a certain level (R)
• Items are ordered multiple times throughout the year
• Occur at any time depending on the demand rate
• System logic dictates:

• Actual quantity ordered

• Continuous monitoring is needed and also known as

• Timing of the order

Perpetual system
• Fixed-time period models are Time triggered in which
Time to reorder is predetermined
• Orders at fixed times quantity depending on demand rate
and therefore monitoring is required periodically.

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Multi-Period Inventory Systems:
Multi-Period Inventory Systems Some Differences
Multi-Period Inventory Models • Fixed-time period model has a larger average inventory
• Fixed-Order Quantity Models
• Fixed-order quantity model favours more expensive items
• Event triggered (Example: running out of stock)
• Fixed-order quantity model more appropriate for important items
• Economic Order Quantity (EOQ Model)
• Q-Model • Fixed-order quantity model require more time to maintain: each
• Fixed-Time Period Models activity is logged
• Time triggered (Example: Monthly sales call by sales
representative)
• Periodic system
• Periodic review system
• Fixed order interval system
• P-Model
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Multi-Period Inventory Systems: Fixed-Order Quantity System

Some Differences
Feature Q-Model P-Model
Demand occurs
Idle state Units withdrawn
Order Quantity Q— constant q— variable Waiting for demand from inventory
or backordered

When to place order R T

Compute inventory position
No Is position ≤
Position = on hand
Recordkeeping Each addition/ At review period reorder point? + on order - backorder
withdrawal
Yes
Inventory size Smaller Larger

Time to maintain Higher Issue an order for

exactly Q units

Type of items A, V, X, H

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Fixed-Time Period Reordering System Fixed-Order Quantity Model:
Model Assumptions
Idle state • Demand for the product is constant and uniform throughout the
Waiting for demand Demand occurs period
Units withdrawn
from inventory
No
or backordered • Lead time (time from ordering to receipt) is constant

Has review • Price per unit of product is constant

time arrived? Yes
Compute inventory position
Position = on hand • Inventory holding cost is based on average inventory
+ on order - backorder

• Ordering or setup costs are constant

Compute order quantity
Issue an order for the
to bring inventory • All demands for the product will be satisfied (No back orders
Number of units needed are allowed)
up to required level

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Basic Fixed-Order Quantity Model and

Inventory Terminology
Reorder Point Behavior

1. You receive an order quantity Q. 4. The cycle then repeats.

TC = Total annual cost
D = Demand
Number
C = Cost per unit of units
Q = Order quantity on hand Q Q Q
S = Cost of placing an order or setup cost
R
H = Annual holding and storage cost per unit of inventory
L L
R = Reorder point 2. You start using
L = Lead time them up over time. 3. When you reach down to a
Time level of inventory of R, you
R = Reorder point
Q = Economic order quantity place your next Q sized order.
L = Lead time

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Inventory Costs Basic EOQ Model

• Typical assumptions made
• Costs associated with ordering too much (represented by • annual demand (D), carrying cost (H) and ordering cost (S) can be
carrying costs) estimated
• Costs associated with ordering too little (represented by • average inventory level is the fixed order quantity (Q) divided by 2

ordering costs) which implies

• no safety stock
• These costs are opposing costs, i.e., as one increases the other
• orders are received all at once
decreases
• demand occurs at a uniform rate
• no inventory when an order arrives
TC = Total annual cost D = Demand • Stockout, customer responsiveness, and other costs are inconsequential
C = Cost per unit Q = Order quantity
• acquisition cost is fixed, i.e., no quantity discounts
S = Cost of placing an order or setup cost
H = Annual holding and storage cost per unit of inventory
R = Reorder point • Annual carrying cost = (average inventory level) x (carrying cost) = (Q/2)H
L = Lead time • Annual ordering cost = (average number of orders per year) x (ordering
cost) = (D/Q)S
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Basic EOQ Model Cost Minimization Goal

• The sum of the two costs is the total stocking cost (TSC) By adding the item, holding, and ordering costs together, we
• When plotted against order quantity, the TSC decreases to a determine the total cost curve, which in turn is used to find the
Qopt inventory order point that minimizes total costs
minimum cost and then increases
• This cost behavior is the basis for answering the first Total Cost

COST
fundamental question: how much to order
Holding
• It is known as the economic order quantity (EOQ) Costs
Annual Cost of
Items (DC)

Ordering Costs

QOPT
Order Quantity (Q)
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Basic Fixed-Order Quantity (EOQ)
Model Formula Deriving the EOQ

Common point on both the cost curves would indicate an

Total Annual Annual Annual optimal point
TC=Total annual cost
Annual = Purchase + Ordering + Holding D =Demand
Cost Cost Cost Cost C =Cost per unit Q  D
Q =Order quantity  H =  S
 
2 Q
D Q S =Cost of placing an
order or setup cost Q2 
TC = DC + S+ H R =Reorder point 
 2 
 H = DS
Q 2 L =Lead time
H=Annual holding and 2 DS
storage cost per unit of Q 2
=
inventory
H

2DS
EOQ =
H

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Deriving the EOQ EOQ Example (1) Problem Data

Using calculus, we take the first derivative of the total
cost function with respect to Q, and set the derivative
Given the information below, what are the EOQ and
(slope) equal to zero, solving for the optimized (cost reorder point?
minimized) value of Qopt
Annual Demand = 1,000 units
2DS 2(Annual Demand)(Order or Setup Cost)
QOPT = = Days per year considered in average
H Annual Holding Cost
daily demand = 365
_ Cost to place an order = $10
We also need a reorder _ Reorder point, R = d L Holding cost per unit per year = $2.50
point to tell us when to d = average daily dem and (co nstant) Lead time = 7 days
place an order L = Lead tim e (constant) Cost per unit = $15
Q OPT 2S
• Optimal cycle length TO = =
D DH
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EOQ Example (1) Solution EOQ Example (2) Problem Data

2D S 2(1,000 )(10) Determine the economic order quantity

Q O PT = = = 89.443 units or 90 units and the reorder point given the following…
H 2.50

1,000 units / year

d = = 2.74 units / day Annual Demand = 10,000 units
365 days / year
Days per year considered in average daily demand = 365
_ Cost to place an order = $10
R eorder point, R = d L = 2.74units / day (7days) = 19.18 or 20 u n its Holding cost per unit per year = 10% of cost per unit
Lead time = 10 days
In summary, you place an optimal order of 90 units. In the Cost per unit = $15
course of using the units to meet demand, when you only
have 20 units left, place the next order of 90 units.

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EOQ Example (2) Solution Example: Basic EOQ

Zartex Co. produces fertilizer to sell to wholesalers. One

2D S 2(10,00 0 )(10) raw material – calcium nitrate – is purchased from a nearby
Q OPT = = = 365.148 units, or 366 u n its
H 1.50 supplier at $22.50 per ton. Zartex estimates it will need
5,750,000 tons of calcium nitrate next year.
10,000 units / year The annual carrying cost for this material is 40% of the
d= = 27.397 units / day
365 days / year acquisition cost, and the ordering cost is $595.
_
R = d L = 27.397 units / day (10 days) = 273.97 or 27 4 u n its a) What is the most economical order quantity?
b) How many orders will be placed per year?
Place an order for 366 units. When in the course of using c) How much time will elapse between orders?
the inventory you are left with only 274 units, place the
next order of 366 units.

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Example: Basic EOQ Example: Basic EOQ

• Economical Order Quantity (EOQ) • Total Annual Stocking Cost (TSC)
D = 5,750,000 tons/year
H = .40(22.50) = $9.00/ton/year TSC = (Q/2)H + (D/Q)S
= (27,573.135/2)(9.00)
S = $595/order
+ (5,750,000/27,573.135)(595)
2DS = 124,079.11 + 124,079.11
EOQ =
H = $248,158.22
2(57,50,000)(595) Note: Total Carrying Cost
EOQ = equals Total Ordering Cost
9.00

= 27,573.135 tons per order

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Example: Basic EOQ Some Practice with EOQ

• Number of Orders Per Year • A trading company buys 6000 units of an item every
= D/Q
year with an unit cost of Rs.300/-. It costs
= 5,750,000/27,573.135
= 208.5 orders/year Rs.1,250/- to process an order and arrange delivery,
while interest and storage costs amount to Rs.60/- a
• Time Between Orders year for each unit held. What is the best ordering
= Q/D Note: This is the inverse policy for the item?
= 1/208.5 of the formula above.
= .004796 years/order
= .004796(365 days/year) = 1.75 days/order

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EOQ Limitations EOQ Limitations

When batch set-up costs are high, EOQ suggests very • EOQ suggests fractional value
large batches. • Suppliers unwilling to split standard package sizes
• Complicates production scheduling • Deliveries by vehicles with fixed capacities
• Give longer lead times to customers
• More convenient to round order size
• Needs excess inventory storage

• Too much capital in stocks

If you shift from EOQ, what happens to the cost?
• Solution?
• Artificially high value on holding cost

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An Example How Robust is the EOQ Model

D = 6000 C = 30 S = 125 H=7 Impact on cost with changes in order quantity near EOQ
2 x 125 x 6000 The Total-Cost Curve is U-Shaped
EOQ = Qo = = 462.91
7 Q D
TC = H+ S

Annual Cost
VCo = H x Q = 3,240.37 2 Q

At 450? VC = DS/Q + QH/2 = 3,241.67

At 500? VC = DS/Q + QH/2 = 3,250.00

For 450 Batch 2.8% below optimal Cost 0.04% High Ordering Costs
For 500 Batch 8% above optimal Cost 0.3% High
QO (optimal order quantity) Order Quantity (Q)

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Example Uncertainty in Demand

Mae Chow Min works in her bakery for 6 days a week for 49 weeks a • Assumption was demand in known
year. Flour is delivered directly with a charge of $7.5 for each
delivery. Chow Min uses an average of 10 sacks of whole-grain flour • Suppose we have erred by E percent
a day, for which she pays $12 a sack. She has an overdraft at the bank • Then demand would be D(1+E)
which costs 12 per cent a year, with spillage, storage, loss and
insurance costing 6.75 per cent a year.
2 S D (1 + E)
A. What size of delivery should Chow Min use Q=
H
and what are the resulting costs?
B. How much should she order if the flour has a
shelf life of 2 weeks? VC/VCo = ½ [Qo/Q + Q/Qo]
C. How much should she order if the bank
imposes a maximum order value of $1,500?
VC 1  1 1+ E 
D. If the mill only delivers on Mondays, how = × + 
much she order and how often? VCo 2  1+ E 1 
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Uncertainty in Costs Adjusting the Order Quantity

• Assumption was costs are known • If uncertain of cost for an item which is already being ordered
regularly.
• Ordering cost and holding cost
• Work backwards to compute ordering cost using EOQ
• Suppose we have taken extra E1 and E2 respectively • Suppose prices are going to go up
• Calculations would then have been for • Marketing promos would increase demand
• Ordering Cost = S(1+E1) • Get EOQs and then increase the order
• Raise EOQ automatically by a factor K
• Holding Cost = H(1+E2)
• Surrogate costs:
• Artificially low holding cost, or
VC/VCo = ½ [Qo/Q + Q/Qo] • High reorder costs

VC 1  1+ E2 1+E1 
= × + 
VCo 2  1+ E1 1+E2 
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Quantity Discounts:
Procedure to Find Best Order Size An Example for Constant Holding Cost
• The maintenance department of a hospital uses about 816
Start 1 Take the next 2 Find the lowest
lowest unit cost point using Qo cases of liquid cleanser annually. Ordering costs are $12,
holding costs are $4 per case a year, and the new price
schedule indicates that orders are less than 50 cases will
Calculate the cost at Is this cost $20 per case, 50 to 79 cases will cost $18 per case, 80
4 the break point on the No
3 point
left of the valid range valid? to 99 cases will cost $17 per case, and larger orders will
cost $16 per case. Determine the optimal order quantity and
Yes the total cost.
Find the lowest cost Compare the costs Calculate the cost
7 and corresponding 6 of all the points 5 at this valid
order size considered minimum

Finish

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Quantity Discounts: Quantity Discounts:

An Example for Proportionate Holding Cost An Example for Proportionate Holding Cost

• Surge Electric uses 4,000 toggle switches a year. Switches

are priced as follows: 1 to 499, 90 cents each; 500 to 999,
85 cents each; and 1000 or more, 80 cents each. It costs
approximately $30 to prepare an order and receive it, and
holding costs 40% of purchase price per unit on an annual
basis. Determine the optimal order quantity and the total
annual cost.

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Economic Production Quantity (EPQ) Economic Production Quantity

• Production done in batches or lots

• Capacity to produce a part exceeds the part’s usage or

Production
Production

& Usage
& Usage
demand rate Usage Usage
• Assumptions of EPQ are similar to EOQ except orders
are received incrementally during production
In
ve
n to
ry
Le
ve
l

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Economic Production Quantity Assumptions Example for Economic Production Quantity

• Only one item is involved • A toy manufacturer uses 48000 rubber-wheels per year
• Annual demand is known for its popular dump truck series. The firm makes its own
wheels, which it can produce at a rate of 800 per day.
• Usage rate is constant The toy trucks are assembled uniformly over the entire
• Usage occurs continually year. Holding cost is $1 per wheel a year. Setup cost for
• Production rate is constant a production run of wheels is $45. the firm operates 240
days per year. Determine the
• Lead time does not vary Economic Run Size a. Optimal run size
• No quantity discounts b. Minimum total annual cost for carrying and setup
c. Cycle time for the optimal run size
2 DS p
Q0 =
d. Run time

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Planned Shortages with Back Orders Stock-Out Occurrence

Shortages are when demand is not met from stock.
Useful when:
• Shortages are not expensive
• Planned shortages are beneficial, e.g., Car dealer, furniture shop
• It is more likely when:
• Unit cost is high
• Wide range of items
• Extreme case is ‘Make-to-Order’
Example Problem–
Demand for an item is constant at 100 units a month. Unit cost is 50,
reorder cost is 50, holding cost is 25% of value a year, shortage cost
for back orders is 40% of value a year. Find an optimal inventory
policy for the item.
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Safety Stock Inventory Level with Safety Stock

Quantity

Maximum probable demand

during lead time

Expected demand
during lead time

Safety stock reduces risk of Safety stock

stockout during lead time L Time

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When to Reorder with EOQ Ordering Reorder Point

• Reorder Point - When the quantity on hand of an item drops
to this amount, the item is reordered
The ROP based on a normal
• Safety Stock - Stock that is held in excess of expected Distribution of lead time demand
demand due to variable demand rate and/or lead time.
• Service Level - Probability that demand will not exceed Service level
supply during lead time. Risk of
a stockout
Probability of
Determinants of the Reorder Point no stockout

• The rate of demand Expected ROP Quantity

• The lead time demand Safety

stock
• Demand and/or lead time variability 0 z z-scale
• Stockout risk (safety stock)

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Basis for Setting the Reorder Point DDLT Distributions

• If there is variability in the DDLT, the DDLT is expressed as a
• The reorder point is set based on distribution
• the demand during lead time (DDLT) and • Discrete (Integer values)
• Continuous (Valid for high demand)
• the desired customer service level

• Reorder point (R) = Expected demand during lead time

(EDDLT) + Safety stock (SS)
Reorder Point for a Discrete DDLT Distribution
• The amount of safety stock needed is based on the degree of
• Assume a probability distribution of actual DDLTs is given or can be
uncertainty in the DDLT and the customer service level developed from a frequency distribution
desired • Starting with the lowest DDLT, accumulate the probabilities. These
are the service levels for DDLTs
• Select the DDLT that will provide the desired customer level as the
reorder point
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Example for Setting Reorder Point Example for Setting Reorder Point

One of Sharp Retailer’s inventory items is now being Construct a Cumulative DDLT Distribution
analyzed to determine an appropriate level of safety stock.
The manager wants an 80% service level during lead time. Probability Probability of
The item’s historical DDLT is:
DDLT (cases) of DDLT DDLT or Less
DDLT (cases) Occurrences
2 0 0
3 8 3 .4 .4
4 6 4 .3 .7
5 4 .8
5 .2 .9
6 2 6 .1 1.0

To provide 80% service level, R = 5 cases

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Example for Setting Reorder Point Continuous DDLT Distribution

Safety Stock (SS)

R = EDDLT + SS
SS = R - EDDLT
EDDLT = .4(3) + .3(4) + .2(5) + .1(6) = 4.0
SS =5–4= 1

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L03-05: Multi Period Inventory Systems SCM 2011
Reorder Point for a
Continuous DDLT Distribution Continuous DDLT Distribution
Z Percentage Cycle
of Cycles service
with shortages Level d = average demand during lead time
(%)
0.00 50.0 50.0
0.84 20.0 80.0 σ d = standard deviation of demand during lead time
1.00 15.9 84.1
1.04 15.0 85.0
1.28 10.0 90.0
1.48 7.0 93.0
1.64 5.0 95.0
1.88 3.0 97.0 Safety Stock = SS = Z σ d
2.00 2.3 97.7
2.33 1.0 99.0
Reorder point = R = d L + Zσ d
2.58 0.5 99.5
3.00 0.1 99.9
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Reorder Point for a Setting Order Point

Continuous DDLT Distribution for a Continuous DDLT Distribution
• The customer service level is converted into a Z value using
• The resulting DDLT distribution is a normal distribution
with the following parameters: the normal distribution table
(Or NORSINV function in excel)
EDDLT = d L • The safety stock is computed by multiplying the Z value by
σDDLT.
• The order point is set using R = EDDLT + SS, or by
σ DDLD = L (σ d ) 2
substitution

_
R = d L + z L (σ d ) 2
Standard deviation of a series of independent occurrences is
equal to the square root of the sum of the variances

σ S = σ 12 + σ 22 + .... + σ i2
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Example for Setting Reorder Point Example for Setting Reorder Point
• EDDLT = 15 units
Auto Zone sells auto parts and supplies including a popular
multi-grade motor oil. When the stock of this oil drops to 20 units, • σDDLT = 6 units
a replenishment order is placed. The store manager is concerned R = EDDLT + Z(σDDLT )
that sales are being lost due to stockouts while waiting for an order. 20 = 15 + Z(6) Standard Normal Distribution
It has been determined that lead time demand is normally 5 = Z(6)
distributed with a mean of 15 units and a standard deviation of 6
Z = 5/6
units. Area = .2967
Z = .833
The manager would like to know the probability of a stockout
during lead time.
Area = .2033

Area = .5 z
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Example for Setting Reorder Point Rules of Thumb in Setting Reorder Point

• The Standard Normal table shows an area of .2967 for the • Set safety stock level at a percentage of EDDLT
region between the z = 0 line and the z = .833 line. The where j is a factor between 0 and 3.
shaded tail area is .5 - .2967 = .2033. Class Description j
OP = EDDLT + j (EDDLT)
1 Uncritical 0.1
2 Uncertain-uncritical 0.2
• The probability of a stockout during lead time is .2033 3 Critical 0.3
4 Uncertain-critical 0.5
5 Supercritical 1.0
6 Uncertain-supercritical 3.0

• Set safety stock level at square root of EDDLT

When stockouts are not
R = EDDLT + EDDLT particularly undesirable

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L03-05: Multi Period Inventory Systems SCM 2011

Fixed Time Period Models Fixed Time Period Benefits

• Tight control of inventory items
• Items from same supplier may yield savings in:
Also known as Fixed Order Interval Model • Ordering
• Orders are placed at fixed time intervals • Packing

• Order quantity for next interval? • Shipping costs

• May be practical when inventories cannot be closely
• Suppliers might encourage fixed intervals monitored
• May require only periodic checks of inventory levels

• Risk of stockout
Fixed Time Period Disadvantages
• Requires a larger safety stock
• Increases holding/carrying cost
• Costs of periodic reviews

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Fixed-Time Period Model

Behavior of Fixed Time Period Systems
with Safety Stock Formula
• As demand for the inventoried item occurs, the inventory level drops q = Average demand + Safety stock – Inventory currently on hand
• When a prescribed period of time has elapsed, the ordering process is
triggered, i.e., the time between orders is fixed or constant q = d (T + L) + Z σ T + L - I
• At that time the order quantity is determined by finding out the
average demand during the vulnerable period plus some safety stock Where :
and subtracting current inventory level on hand plus on order if any. q = quantitiy to be ordered
• After the lead time elapses, the ordered quantity is received , and the T = the number of days between reviews
inventory level increases L = lead time in days
• The upper inventory level may be determined by the amount of space d = forecast average daily demand
allocated to an item z = the number of standard deviations for a specified service probabilit y
• This system is used where it is desirable to physically count inventory σ T + L = standard deviation of demand over the review and lead time
each time an order is placed I = current inventory level (includes items on order)

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L03-05: Multi Period Inventory Systems SCM 2011
Fixed-Time Period Model:
Determining Quantity in Fixed period Model Determining the Value of σT+L
• Using an approach similar to that used to derive EOQ, the

∑ (σ )
optimal value of the fixed time between orders is derived to T+L

be σ T+L = di
2

i =1

Fixed Period = EOP = 2S/DH

Since each day is independen t and σ d is constant,
σ T + L = (T + L)σ d 2
The standard deviation of a sequence of random
events equals the square root of the sum of the
variances.

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Example of the Fixed-Time Period Model:

Example of the Fixed-Time Period Model Solution (Part 1)
Given the information below, how many units
σ T+ L = (T + L)σ d 2 = ( 30 + 10 )( 4 ) 2 = 25.298
should be ordered?

Average daily demand for a product is 20 units. The review The value for “z” is found by using the Excel
period is 30 days, and lead time is 10 days. Management NORMSINV function.
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 daily demand standard
deviation is 4 units. For a probability 0.96, z = 1.75

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Example of the Fixed-Time Period Model:
Solution (Part 2) Hybrid Inventory Models

• Optional replenishment model

q = d(T + L) + Z σ T + L - I • Similar to the fixed order period model
• Unless inventory has dropped below a prescribed
q = 20(30 + 10) + (1.75)(25.298) - 200 level when the order period has elapsed, no order is
placed
q = 800 + 44.272 - 200 = 644.272, or 645 units • Protects against placing very small orders

• Attractive when review and ordering costs are large

So, to satisfy 96 percent of the demand,

you should place an order of 645 units at
this review period
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Hybrid Inventory Models: Hybrid Inventory Models:

Optional Replenishment System Base Stock Model
• Start with a certain inventory level
Maximum Inventory Level, M
• Whenever a withdrawal is made, an order of equal size is
placed
q=M-I
• Ensures that inventory maintained at an approximately

Actual Inventory Level, I

constant level
M • Appropriate for very expensive items with small ordering
I costs

Q = minimum acceptable order quantity

If q > Q, order q, otherwise do not order any.

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Hybrid Inventory Models: Hybrid Inventory Models:
Single Bin System Two-Bin System

Essentially a P system Essentially a Q system

• When the first bin is empty, it triggers the
• Target inventory level and current inventory position
IP are established replenishment order
• The second bin contains an amount equal to
safety stock,
or the average demand during the lead time plus
the safety stock.
Order Enough to
Refill Bin
Order One Bin of
Inventory
Full Empty
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