1 Inventory Management and Control. 2AMAZON.com Jeff Bezos, in 1995, started AMAZON.com as a...

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Inventory Inventory Management Management andand

ControlControl

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AMAZON.comAMAZON.com

• Jeff Bezos, in 1995, started AMAZON.com as a “virtual” retailer – no inventory, no warehouses, no overhead; just a bunch of computers.

• Growth forced AMAZON.com to excel in inventory management!

• AMAZON is now a worldwide leader in warehouse management and automation.

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Order Fulfillment at AMAZOOrder Fulfillment at AMAZON (1 of 2)N (1 of 2)

1. You order items; computer assigns your order to distribution center [closest facility that has the product(s)]

2. Lights indicate products ordered to workers who retrieve product and reset light.

3. Items placed in crate with items from other orders, and crate is placed on conveyor. Bar code on item is scanned 15 times – virtually eliminating error.

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Order Fulfillment at AMAZONOrder Fulfillment at AMAZON (2 of 2) (2 of 2)

4. Crates arrive at a central point where items are boxed and labeled with new bar code.

5. Gift wrapping done by hand (30 packages per hour)

6. Box is packed, taped, weighed and labeled before leaving warehouse in a truck.

7. Order appears on your doorstep within a week

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InventoryInventory ( (DefinDefinition of)ition of)

• Inventory is the stock of any item or resource held to meet future demand and can include: raw materials, finished products, component parts, supplies, and work-in-process

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Inventory

Process stage

Demand Type

Number & Value Other

Raw Material WIP

Finished Goods

Independent Dependent

A Items B Items C Items

Maintenance Operating

Inventory ClassificationsInventory Classifications

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E(1)

Inventories by Demand TypeInventories by Demand Type

B(4)

E(2)D(1)

C(2)

E(3)B(1)

A

Independent Demand : Demand for the final end-product that are ready to be sold or used. Demand not related to other items; demand created by external customers); eg. Demand for computers

Dependent Demand : Derived demand for components of finished products (parts, raw materials, subassemblies)

Finished product: eg: Computer

Components: eg. parts that make up the computer

Independent demand is uncertain Dependent demand is certain

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Types of InventoriesTypes of Inventories (1 of 2) (1 of 2)

• Raw materials & purchased parts

• Partially completed goods called work in process

• Finished-goods inventories (manufacturing firms) or merchandise (retail stores)

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Types of Inventories (Types of Inventories (2 of 22 of 2))

• Maintenance and repairs (MRO) inventory, replacement parts, tools, & supplies

• Goods-in-transit to warehouses or customers (pipeline inventory)

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The Material Flow CycleThe Material Flow Cycle (1 of 2) (1 of 2)

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Run time: Job is at machine and being worked onSetup time: Job is at the work station, and the work station is

being "setup."Queue time: Job is where it should be, but is not being

processed because other work precedes it.Move time: The time a job spends in transitWait time: When one process is finished, but the job is waiting

to be moved to the next work area.Other: "Just-in-case" inventory.

The Material Flow CycleThe Material Flow Cycle (2 of 2) (2 of 2)

WaitTime

MoveTime

QueueTime

SetupTime

RunTimeInput

Cycle Time

Output

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Performance MeasuresPerformance Measures

• Inventory turnover (the ratio of annual cost of goods sold to average inventory investment)

• Days of inventory on hand (expected number of days of sales that can be supplied from existing inventory)

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Functions Functions of Inventorof Inventories (1 of 2)ies (1 of 2)

1. To meet variation in product demand and to protect against stock-outs

2. To “decouple” operations or separate various parts of the production process, ie. to maintain independence of operations

3. To meet unexpected demand & to provide high levels of customer service

3. To smooth production requirements by meeting seasonal or cyclical variations in demand

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Functions Functions of Inventorof Inventories (2 of 2)ies (2 of 2)

4. To provide a safeguard for variation in raw material delivery time

5. To provide a stock of goods that will provide a “selection” for customers

6. To take advantage of economic purchase-order size

7. To take advantage of quantity discounts

8. To take advantage of order cycles

9. To hedge against price increases

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• Higher costs– Item cost (if purchased)– Ordering (or setup) cost– Holding (or carrying) cost

• Difficult to control• Hides production problems• May decrease flexibility

Disadvantages of InventorDisadvantages of Inventoriesies

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Inventory CostsInventory Costs Holding (or carrying) costs

Costs for storage, handling, insurance, etc

Setup (or production change) costsCosts to prepare a machine or process for

manufacturing an order, eg. arranging specific equipment setups, etc

Ordering costs (costs of replenishing inventory)Costs of placing an order and receiving goods

Shortage costsCosts incurred when demand exceeds supply

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Holding (Carrying) CostsHolding (Carrying) Costs

• Obsolescence• Insurance• Extra staffing• Interest• Pilferage• Damage• Warehousing• Etc.

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Inventory Holding CostsInventory Holding Costs(Approximate Ranges)(Approximate Ranges)

Category

Housing costs (building rent, depreciation, operating cost, taxes, insurance)

Material handling costs (equipment, lease or depreciation, power, operating cost)

Labor cost from extra handlingInvestment costs (borrowing costs, taxes,

and insurance on inventory)

Pilferage, scrap, and obsolescence

Overall carrying cost

Cost as a % of Inventory Value

6%(3 - 10%)

3%(1 - 3.5%)

3%(3 - 5%)

11%(6 - 24%)

3% (2 - 5%)

26%

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Ordering CostsOrdering Costs

• Supplies• Forms• Order processing• Clerical support, etc.

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Setup CostsSetup Costs

• Clean-up costs• Re-tooling costs• Adjustment costs, etc.

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Shortage CostsShortage Costs

• Backordering cost• Cost of lost sales

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Inventory Control System Inventory Control System DefinedDefined

An inventory system is the set of policies and controls that monitor levels of inventory and determine what levels should be maintained, when stock should be replenished and how large orders should be

Management has two basic functions concerning inventory:Establish a system for tracking items in inventoryMake decisions about: When to order? How much to order?

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Objective of Inventory ControlObjective of Inventory Control

To achieve satisfactory levels of customer service while keeping inventory costs within reasonable bounds

Level of customer service

Costs of ordering and carrying inventory

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A system to keep track of inventory

A reliable forecast of demand

Knowledge of lead time and lead time variability

Reasonable estimates of

Holding costs

Ordering costs

Shortage costs

A classification system for inventory items

Requirements of an Requirements of an Effective Inventory Effective Inventory ManagementManagement

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Inventory CountingInventory Counting (Control) (Control) Systems Systems• Periodic System

Physical count of items made at periodic intervals; order is placed for a variable amount after fixed passage of time

• Perpetual (Continuous) Inventory System System that keeps track of removals from inventory continuously, thus monitoring current levels of each item (constant amount is ordered when inventory declines to a predetermined level)

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• Inventory accuracy refers to how well the inventory records agree with physical count.

• Cycle Counting

Physically counting a sample of total inventory on a regular basis

• Used often with ABC classification– A items counted most often (e.g., daily)

Inventory Accuracy and Inventory Accuracy and Cycle CountingCycle Counting

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Cycle CountingCycle Counting Management Management

• Cycle counting management– How much accuracy is needed?

• A items: ± 0.2 percent

• B items: ± 1 percent

• C items: ± 5 percent

– When should cycle counting be performed?

– Who should do it?

12-27

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Advantages of Cycle CountingAdvantages of Cycle Counting

• Eliminates shutdown and interruption of production necessary for annual physical inventories

• Eliminates annual inventory adjustments• Provides trained personnel to audit the accuracy of

inventory• Allows the cause of errors to be identified and

remedial action to be taken• Maintains accurate inventory records

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Inventory Counting TechnologiesInventory Counting Technologies

• Universal product code (UPC)– Bar code printed on a label that has information

about the item to which it is attached• Radio frequency identification (RFID) tags

– A technology that uses radio waves to identify objects, such as goods in supply chains

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Demand Forecasts and Lead TimeDemand Forecasts and Lead Time

• Forecasts– Inventories are necessary to satisfy customer demands, so

it is important to have a reliable estimates of the amount and timing of demand

• Lead time– Time interval between ordering and receiving the order

• Point-of-sale (POS) systems– A system that electronically records actual sales

– Such demand information is very useful for enhancing forecasting and inventory management

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ABC Classification System ABC Classification System

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ABC Classification SystemABC Classification System

• Demand volume and value of items vary

• Items kept in inventory are not of equal importance in terms of:

– dollars invested

– profit potential

– sales or usage volume

– stock-out penalties

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ABC Classification SystemABC Classification System

Classifying inventory according to some measure of importance and allocating control efforts accordingly.

AA - very important (10 to 20 percent of the number of items in inventory and about 60 to 70 percent of the annual dollar valueBB - mod. importantCC - least important (50 to 60 percent of the number of items in inventory but only about 10 to 15 percent of the annual dolar value

Annual $ value of items

High

LowFew ManyNumber of Items

AA

CCBB

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Classify inventory into 3 categories typically on the basis of the dollar value to the firm $ volume = Annual demand x Unit cost

A class, B class, C class Policies based on ABC analysis– Develop class A suppliers more carefully – Give tighter physical control of A items– Forecast A items more carefully

ABC AnalysisABC Analysis

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% of Inventory Items

Classifying Items as ABCClassifying Items as ABC

0

20

40

60

80

100

50 100

% Annual $ Usage

AABB

CC

Class % $ Vol % ItemsA 70-80 5-15B 15 30C 5-10 50-60

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ABC ClassificationABC Classification

1 $ 60 902 350 403 30 1304 80 605 30 1006 20 1807 10 1708 320 509 510 60

10 20 120

PART UNIT COST ANNUAL USAGE

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11 $ 60$ 60 909022 350350 404033 3030 13013044 8080 606055 3030 10010066 2020 18018077 1010 17017088 320320 505099 510510 6060

1010 2020 120120

PARTPART UNIT COSTUNIT COST ANNUAL USAGEANNUAL USAGETOTAL % OF TOTAL % OF TOTALPART VALUE VALUE QUANTITY % CUMMULATIVE

9 $30,600 35.9 6.0 6.08 16,000 18.7 5.0 11.02 14,000 16.4 4.0 15.01 5,400 6.3 9.0 24.04 4,800 5.6 6.0 30.03 3,900 4.6 10.0 40.06 3,600 4.2 18.0 58.05 3,000 3.5 13.0 71.0

10 2,400 2.8 12.0 83.07 1,700 2.0 17.0 100.0

$85,400

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11 $ 60$ 60 909022 350350 404033 3030 13013044 8080 606055 3030 10010066 2020 18018077 1010 17017088 320320 505099 510510 6060

1010 2020 120120

PARTPART UNIT COSTUNIT COST ANNUAL USAGEANNUAL USAGETOTAL % OF TOTAL % OF TOTALPART VALUE VALUE QUANTITY % CUMULATIVE

9 $30,600 35.9 6.0 6.08 16,000 18.7 5.0 11.02 14,000 16.4 4.0 15.01 5,400 6.3 9.0 24.04 4,800 5.6 6.0 30.03 3,900 4.6 10.0 40.06 3,600 4.2 18.0 58.05 3,000 3.5 13.0 71.0

10 2,400 2.8 12.0 83.07 1,700 2.0 17.0 100.0

$85,400

AA

BB

CC

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11 $ 60$ 60 909022 350350 404033 3030 13013044 8080 606055 3030 10010066 2020 18018077 1010 17017088 320320 505099 510510 6060

1010 2020 120120

PARTPART UNIT COSTUNIT COST ANNUAL USAGEANNUAL USAGETOTAL % OF TOTAL % OF TOTALPART VALUE VALUE QUANTITY % CUMMULATIVE

9 $30,600 35.9 6.0 6.08 16,000 18.7 5.0 11.02 14,000 16.4 4.0 15.01 5,400 6.3 9.0 24.04 4,800 5.6 6.0 30.03 3,900 4.6 10.0 40.06 3,600 4.2 18.0 58.05 3,000 3.5 13.0 71.0

10 2,400 2.8 12.0 83.07 1,700 2.0 17.0 100.0

$85,400

AA

BB

CC

% OF TOTAL % OF TOTALCLASS ITEMS VALUE QUANTITY

A 9, 8, 2 71.0 15.0B 1, 4, 3 16.5 25.0C 6, 5, 10, 7 12.5 60.0

40


100 100 –

80 80 –

60 60 –

40 40 –

20 20 –

0 0 –| | | | | |00 2020 4040 6060 8080 100100

% % of Quantityof Quantity

% o

f V

alu

e%

of

Val

ue

AA

BBCC

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Inventory Inventory ModelsModels Multi-Period Inventory Models

Fixed-Order Quantity Models• Event triggered (Example: running out of stock)

Fixed-Time Period Models • Time triggered (Example: Monthly sales call by

sales representative) Single-Period Inventory Model

One time purchasing decision (Example: vendor selling t-shirts at a football game)

Seeks to balance the costs of inventory overstock and under stock

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Multi-Period Inventory Models Multi-Period Inventory Models

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Multi-Period Inventory ModelsMulti-Period Inventory Models

Fixed-Order Quantity Models (Types of)The Basic Economic Order Quantity ModelEconomic Production Order Quantity (Economic Lot Size) ModelEconomic Order Quantity Model with Quantity Discounts

Fixed Time Period (Fixed Order Interval) Models

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Fixed Order Quantity Models:Economic Order Quantity Model

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Economic Economic Order QuantiOrder Quantity Modelty Model

The basic EOQ Model is used to find a fixed order quantity that will minimize total annual inventory costs

Assumptions:

• Only one product is involved

• Demand for the product is known with certainty, is constant and uniform (even) throughout the period

• Lead time (time from ordering to receipt) is known and constant

• Price per unit of product is constant (no quantity discounts)

• Inventory holding cost is based on average inventory

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Economic OrderEconomic Order Quantity Model Quantity Model

• Ordering or setup costs are constant

• All demands for the product will be satisfied (no back orders are allowed)

• No stockouts (shortages) are allowed

• The order quantity is received all at once. (Instantaneous receipt of material in a single lot)

The goal is to calculate the order quantitiy that minimizes total cost

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Basic Fixed-Order Quantity Model and Basic Fixed-Order Quantity Model and Reorder Point BehaviorReorder Point Behavior

R = Reorder pointQ = Economic order quantityL = Lead time

L L

Q QQ

R

Numberof unitson hand(Inv. Level)

1. You receive an order quantity Q.

2. You start using them up over time.(usage rate) 3. When you reach down to a level

of inventory of R, you place your next Q sized order.

4. The cycle then repeats.

Place order

Receive order

Time

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EOQ ModelEOQ Model

Reorder Point

(ROP)

Time

Inventory LevelAverage

Inventory (Q/2)

Lead Time

Order Quantity

(Q)

Demand rate

Order placed Order received

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Total Annual Cost Total Annual Cost

H 2

Q + S

Q

D + DC = TC H

2

Q + S

Q

D + DC = TC

Total Annual =Cost

AnnualPurchase

Cost

AnnualOrdering

Cost

AnnualHolding

Cost+ +

TC=Total annual costD =Annual demandC =Cost per unitQ =Order quantity in unitsS =Cost of placing an order or setup costR =Reorder pointL =Lead timeH=Annual holding (carrying) and storage cost per unit of inventory

TC=Total annual costD =Annual demandC =Cost per unitQ =Order quantity in unitsS =Cost of placing an order or setup costR =Reorder pointL =Lead timeH=Annual holding (carrying) and storage cost per unit of inventory

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EOQ EOQ Cost Cost ModelModel: H: How Much to Order?ow Much to Order?By adding the holding and ordering costs together, we determine the total cost curve, which in turn is used to find the optimal order quantity that minimizes total costs

Slope = 0Slope = 0

Total CostTotal Cost

Order Quantity, Order Quantity, QQ

Annual Annual cost ($)cost ($)

Minimum Minimum total costtotal cost

Optimal orderOptimal order QQoptopt

Carrying Cost =Carrying Cost =HHQQ

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Ordering Cost =Ordering Cost =SSDD

QQ

SQ

DH

QTC

2

The total cost curve is U-Shaped

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• More units must be stored if more are ordered

Purchase OrderDescription Qty.Microwave 1

Order quantity


Order quantity

Why Holding Costs IncreaseWhy Holding Costs Increase??

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Cost is spread over more units

Example: You need 1000 microwave ovens





1 Order (Postage $ 0.33) 1000 Orders (Postage $330)

Order quantity

Purchase Order

Description Qty.Microwave 1000

Why OrderWhy Orderinging Costs Decrease Costs Decrease ? ?

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Basic Fixed-Order Quantity (EOQ) Model Basic Fixed-Order Quantity (EOQ) Model FormulaFormula

H 2

Q + S

Q

D + DC = TC H

2

Q + S

Q

D + DC = TC

Total Annual =Cost

AnnualPurchase

Cost

AnnualOrdering

Cost

AnnualHolding

Cost+ +

TC=Total annual costD =Annual demandC =Cost per unitQ =Order quantityS =Cost of placing an order or setup costR =Reorder pointL =Lead timeH=Annual holding and storage cost per unit of inventory

TC=Total annual costD =Annual demandC =Cost per unitQ =Order quantityS =Cost of placing an order or setup costR =Reorder pointL =Lead timeH=Annual holding and storage cost per unit of inventory

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EOQ Cost ModelEOQ Cost Model

Annual ordering cost =Annual ordering cost =S S DD

QQ

AnnualAnnual carrying costcarrying cost ==HHQQ

22

Total cost = +Total cost = +S S DD

QQH H QQ

22

TC = +S D

Q

H Q

2

= +S D

Q2

H

2TC

Q

0 = +S D

Q2

H

2

Qopt =2SD

H

Deriving Qopt

Using calculus, we take the first derivative of the total cost function with respect to Q, and set the derivative (slope) equal to zero, solving for the optimized (cost minimized) value of Qopt . The total cost curve reaches its minimum where the carrying and ordering costs are equal

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Deriving the EOQDeriving the EOQ

Q = 2DS

H =

2(Annual D em and)(Order or Setup Cost)

Annual Holding CostOPTQ =

2DS

H =

2(Annual D em and)(Order or Setup Cost)

Annual Holding CostOPT

Reorder point, R = d L_

Reorder point, R = d L_

d = average daily demand (constant)

L = Lead time (constant)

_We also need a reorder point to tell us when to place an order

We also need a reorder point to tell us when to place an order

How much to order?:

When to order?

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Expected Number of Orders

Expected Time Between Orders Working Days / Year

Working Days / Year

= =

= =

=

= ×

NDQ*

TN

dD

ROP d L

EOQ Model EquationsEOQ Model Equations

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EOQ ExampleEOQ Example 1 (1 of 3) 1 (1 of 3)

Annual Demand = 1,000 unitsDays per year considered in average daily demand = 365Cost to place an order = $10Holding cost per unit per year = $2.50Lead time = 7 daysCost per unit = $15

Given the information below, what are the EOQ and reorder point?

Given the information below, what are the EOQ and reorder point?

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EOQ Example EOQ Example 11((2 of 32 of 3))

Q = 2DS

H =

2(1,000 )(10)

2.50 = 89.443 units or OPT 90 unitsQ =

2DS

H =

2(1,000 )(10)

2.50 = 89.443 units or OPT 90 units

d = 1,000 units / year

365 days / year = 2.74 units / dayd =

1,000 units / year

365 days / year = 2.74 units / day

Reorder point, R = d L = 2.74units / day (7days) = 19.18 or _

20 units Reorder point, R = d L = 2.74units / day (7days) = 19.18 or _

20 units

In summary, you place an optimal order of 90 units. In the course of using the units to meet demand, when you only have 20 units left, place the next order of 90 units.

In summary, you place an optimal order of 90 units. In the 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 ExampleEOQ Example 1(3 of 3) 1(3 of 3)

TCmin =SD

Q

HQ

2

TCmin = (10)(1,000)

90

(2,5)(90)

2

TCmin = $ 111 + $111 = 222 $

Orders per year = D/Qopt

= 1000/90

= 11 orders/year

Order cycle time = 365/(D/Qopt)

= 365/11 = 33.1days

+

+

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EOQ Example EOQ Example 22((1 of 1 of 2)2)

Annual Demand = 10,000 unitsDays per year considered in average daily demand = 365Cost to place an order = $10Holding cost per unit per year = 10% of cost per unitLead time = 10 daysCost per unit = $15

Determine the economic order quantity and the reorder point given the following…

Determine the economic order quantity and the reorder point given the following…

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EOQ Example EOQ Example 22(2(2 of 2 of 2))

Q =2DS

H=

2(10,000 )(10)

1.50= 365.148 units, or OPT 366 unitsQ =

2DS

H=

2(10,000 )(10)

1.50= 365.148 units, or OPT 366 units

d =10,000 units / year

365 days / year= 27.397 units / dayd =

10,000 units / year

365 days / year= 27.397 units / day

R = d L = 27.397 units / day (10 days) = 273.97 or _

274 unitsR = d L = 27.397 units / day (10 days) = 273.97 or _

274 units

Place an order for 366 units. When in the course of using the inventory you are left with only 274 units, place the next order of 366 units.

Place an order for 366 units. When in the course of using the inventory you are left with only 274 units, place the next order of 366 units.

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EOQ ExampleEOQ Example 3 3

H = $0.75 per yard S = $150 D = 10,000 yards

Qopt =2 S D

H

Qopt =2(150)(10,000)

(0.75)

Qopt = 2,000 yards

TCmin = +S D

Q

H Q

2

TCmin = +(150)(10,000)

2,000

(0.75)(2,000)

2

TCmin = $750 + $750 = $1,500

Orders per year = D/Qopt

= 10,000/2,000

= 5 orders/year

Order cycle time =311 days/(D/Qopt)

= 311/5

= 62.2 store days

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When to Reorder with EOQ OrderingWhen to Reorder with EOQ Ordering ? ?Reorder Point

–When the quantity on hand of an item drops to this amount, the item is reordered

ROP = d . Lwhere:d= demand rate (units per period, per day, per week)L= lead time (in the same unts as d)

–Determinants of the reorder point1. the rate of demand2. the lead time3. the extent of demand and/or lead time variability4. the degree of stockout risk acceptable to management

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Reorder Point ExampleReorder Point Example

Demand = 10,000 yards/year

Store open 311 days/year

Daily demand = 10,000 / 311 = 32.154 yards/day

Lead time = L = 10 days

R = dL = (32.154)(10) = 321.54 yards

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Reorder Point: Under UncertaintyReorder Point: Under Uncertainty

StockSafety timelead during

demand Expected ROP

12-65

• Demand or lead time uncertainty creates the possibility that demand will be greater than available supply• To reduce the likelihood of a stockout, it becomes necessary to carry safety stock Safety Stock • Stock that is held in excess of expected demand due to variable demand and/or lead time

66

Safety StockSafety Stock

LT Time

Expected demandduring lead time

Maximum probable demandduring lead time

ROP

Qu

an

tity

Safety stock

Safety stock reduces risk ofstockout during lead time

67

Variable Demand with Variable Demand with a Reorder Pointa Reorder Point

Reorderpoint, R

Q

LTLT

TimeTimeLTLT

Inve

nto

ry le

vel

0

68

Reorder Point withReorder Point with a Safety Stocka Safety Stock

Reorderpoint, R

QQ

LT

Time

LT

Inve

nto

ry le

vel

0

Safety Stock

69

Safety Stock?Safety Stock?• As the amount of safety stock carried increases, the risk of

stockout decreases.– This improves customer service level

• Service level– The probability that demand will not exceed supply during

lead time (probability that inventory available during lead time will meet demand)

– Service level = 100% - Stockout risk (probability of stockout)

- Higher service level means more safety stock

- More safety stock means higher ROP12-69

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How Much Safety Stock?How Much Safety Stock?• The amount of safety stock that is appropriate for a

given situation depends upon:

1. The average demand rate and average lead time

2. Demand and lead time variability

3. The desired service level

demand timelead ofdeviation standard The

deviations standard ofNumber

where

timelead during

demand Expected R

dLT

dLT

z

z

12-70

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Reorder Point for Service LevelReorder Point for Service Level

Probability of a stockout

R

Safety stock

dLExpected Demand

zd L

The reorder point based on a normal distribution of LT demand

72

Reorder PointReorder Point

ROP

Probability of a stockoutService level

Expecteddemand

Safety stock

0 z

Quantity

z-scale

12-72

Probability of meeting demand during lead time (Probability of no stockout)= service level

The ROP based on a normal distribution of lead time demand

zd √ L

73

Reorder Point With Variable DemandReorder Point With Variable Demand

R = dL + zd L

L :Note ddLT

where

d = average daily demandL = lead time (same time units as average demand)d = the standard deviation of daily demand(same time units as average demand)

z = number of standard deviations corresponding to the service levelprobability

zd L = safety stock

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Reorder Point for Reorder Point for V Variable Demandariable Demand (Example(Example))

The carpet store wants a reorder point with a 95% service level and a 5% stockout probability

d = 30 yards per dayL = 10 daysd = 5 yards per day

For a 95% service level, z = 1.65

R = dL + z d L

= 30(10) + (1.65)(5)( 10)

= 326.1 yards

Safety stock = z d L

= (1.65)(5)( 10)

= 26.1 yards

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Reorder Point: Lead Time UncertaintyReorder Point: Lead Time Uncertainty

) as units time(same timelead Average LT

) as units time(same timelead of stddev. The

per week) day,(per periodper Demand

deviations standard ofNumber

where

LT ROP

LT

LT

d

d

d

z

zdd

12-75

76

Fixed Order Quantity Models:Fixed Order Quantity Models:-Noninstantaneous Receipt--Noninstantaneous Receipt-Production Order Quantity Production Order Quantity

(Economic Lot Size) (Economic Lot Size) ModelModel

77

Economic Production Quantity (EPQ)Economic Production Quantity (EPQ) or Economic Order Quantityor Economic Order Quantity

or Economic Lot Sizeor Economic Lot Size

• Assumptions– Only one product is involved

– Annual demand requirements are known

– Usage rate is constant

– Usage occurs continually, but production occurs periodically

– The production rate is constant

– Lead time does not vary

– There are no quantity discounts

12-77

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Production done in batches or lotsCapacity to produce a part exceeds that part’s

usage or demand rateAllows partial receipt of material

Other EOQ assumptions apply

Suited for production environment Material produced, used immediately Provides production lot size

Lower holding cost than EOQ modelAnswers how much to order and when to order

Production Order Quantity ModelProduction Order Quantity Model

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EOQ EOQ EPQ: Inventory ProfileEPQ: Inventory Profile Q

Q*

Imax

Productionand usage

Productionand usage

Productionand usage

Usageonly

Usageonly

Cumulativeproduction

Amounton hand

Time

12-79

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EOQ POQ ModelEOQ POQ ModelWhen To OrderWhen To Order

Reorder Point (ROP)

Time

Inventory Level

AverageInventory

Lead Time

Optimal Order Quantity(Q*)

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POQ Model Inventory LevelsPOQ Model Inventory Levels

Time

Inventory Level

Production Portion of

Cycle

Max. Inventory Level Q·(1- u/p)Q*Q*

Supply Begins

Supply Ends

Inventory level with no demand

Demand portion of cycle with no supply

Average inventory (Q/2)(1- u/p)

82

EPQ – Total CostEPQ – Total Cost

rate Usage

ratedelivery or Production

inventory Maximum

where

2

Cost SetupCost CarryingTC

max

max

u

p

upp

Q

I

SQ

DH

I

12-82

83

POQ Model EquationsPOQ Model Equations

Production Order Quantity = =

-

Q

H* up

p*

1

2*D*S

( )

QDS

H

p

p u0

2

84

Production Production Order Order QuantityQuantity Example Example (1 of 2)(1 of 2)


u = 10,000/311 = 32.2 yards per day p = 150 yards per day

POQopt= = = 2,256.8 yards

2 S D

H 1 - up

2(150)(10,000)

0.75 1 - 32.2150

TC = + 1 - = $1,329up

S DQ

H Q2

Production run = = = 15.05 days per orderQp

2,256.8150

85

Production QuantityProduction Quantity Example Example (2 of 2)(2 of 2)


u= 10,000/311 = 32.2 yards per day p = 150 yards per day

QQoptopt = = = 2,256.8 yards = = = 2,256.8 yards

22CCooDD

CCcc 1 - 1 - ddpp

2(150)(10,000)2(150)(10,000)

0.75 1 - 0.75 1 - 32.232.2150150

TCTC = + 1 - = $1,329 = + 1 - = $1,329ddpp

CCooDD

QQ

CCccQQ

22

Production run = = = 15.05 days per orderProduction run = = = 15.05 days per orderQQpp

2,256.82,256.8150150

86

Number of production runs = = = 4.43 runs/yearDQ

10,0002,256.8

Maximum inventory level = Q (1 - ) = 2,256.8 ( 1 -

= 1,772 yards

up

32.2 150

Production QuantityProduction Quantity Example Example (2 of 2)(2 of 2)

87

Fixed-Order Quantity Models:Fixed-Order Quantity Models:Economic Order Quantity Model Economic Order Quantity Model

with Quantity Discountswith Quantity Discounts

88

• Answers how much to order & when to order• Allows quantity discounts

– Price reduction of fered to customers for placing large orders, ie. Price per unit decreases as order quantity increases

– Other EOQ assumptions apply• Trade-off is between lower price & increased

holding cost

Quantity Discount ModelQuantity Discount Model

TC = + H + PDS D

Q

Q

2Where P: Unit Price

Total cost with purchasing cost

89

Price-Break Model FormulaPrice-Break Model Formula

Cost Holding Annual

Cost) Setupor der Demand)(Or 2(Annual =

iC

2DS = QOPT

Based on the same assumptions as the EOQ model, the price-break model has a similar Qopt formula:

i = percentage of unit cost attributed to carrying inventoryC = cost per unit

Since “C” changes for each price-break, the formula above will have to be used with each price-break cost value

90

Total Costs with PDTotal Costs with PDC

ost

EOQ

TC with PD

TC without PD

PD

0 Quantity

Adding Purchasing costdoesn’t change EOQ

91

Total Cost with Constant Carrying CostsTotal Cost with Constant Carrying Costs

OC

EOQ Quantity

Tot

al C

ost TCa

TCc

TCbDecreasing Price

CC a,b,c

92

Quantity DiscountsQuantity Discounts

12-92

93

Quantity Discount ScheduleQuantity Discount Schedule

Discount Number

Discount Quantity

Discount (%)

Discount Price (P)

1 0 to 999 No discount $5.00

2 1,000 to 1,999 4 $4.80

3 2,000 and over 5 $4.75

94

Quantity Discount – How Much to OrderQuantity Discount – How Much to Order??

95

Price-Break Example 1 (1 of 3)Price-Break Example 1 (1 of 3)

ORDER SIZE PRICE

0 - 99 $10

100 - 199 8 (d1)

200+ 6 (d2)

For this problem holding cost is given as a constant value, not as a percentage of price, so the optimal order quantity is the same for each of the price ranges. (see the figure)

96

Price Break Example 1 (2 of 3)Price Break Example 1 (2 of 3)

QQoptopt

Carrying cost Carrying cost

Ordering cost Ordering cost

Inve

nto

ry c

ost

($)

Inve

nto

ry c

ost

($)

QQ((dd1 1 ) = 100) = 100 QQ((dd2 2 ) = 200) = 200

TC TC ((dd2 2 = $6 ) = $6 )

TCTC ( (dd1 1 = $8 )= $8 )

TC TC = ($10 )= ($10 )

97

Price Break Example 1 (3 of 3)Price Break Example 1 (3 of 3)

QQoptopt

Carrying cost Carrying cost

Ordering cost Ordering cost

Inve

nto

ry c

ost

($)

Inve

nto

ry c

ost

($)

QQ((dd1 1 ) = 100) = 100 QQ((dd2 2 ) = 200) = 200

TC TC ((dd2 2 = $6 ) = $6 )

TCTC ( (dd1 1 = $8 )= $8 )

TC TC = ($10 )= ($10 )

The lowest total cost is at the second price break

98

Price Break Example 2Price Break Example 2

QUANTITYQUANTITY PRICEPRICE

1 - 491 - 49 $1,400$1,400

50 - 8950 - 89 1,1001,100

90+90+ 900900

SS = = $2,500 $2,500

HH = = $190 per computer $190 per computer

DD = = 200200

QQoptopt = = = 72.5 PCs = = = 72.5 PCs22SSDD

HH2(2500)(200)2(2500)(200)

190190

TCTC = + + = + + PD PD = $233,784 = $233,784 SSDD

QQoptopt

H H QQoptopt

22

For For QQ = 72.5 = 72.5

TCTC = + + = + + PD PD = $194,105= $194,105SSDD

QQ

H H QQ

22

For For QQ = 90 = 90

99

Price-Break Example Price-Break Example 33 (1(1 of 4 of 4))

A company has a chance to reduce their inventory ordering costs by placing larger quantity orders using the price-break order quantity schedule below. What should their optimal order quantity be if this company purchases this single inventory item with an e-mail ordering cost of $4, a carrying cost rate of 2% of the inventory cost of the item, and an annual demand of 10,000 units?

A company has a chance to reduce their inventory ordering costs by placing larger quantity orders using the price-break order quantity schedule below. What should their optimal order quantity be if this company purchases this single inventory item with an e-mail ordering cost of $4, a carrying cost rate of 2% of the inventory cost of the item, and an annual demand of 10,000 units?

Order Quantity(units) Price/unit($)0 to 2,499 $1.202,500 to 3,999 1.004,000 or more .98

100

Price-Break ExamplePrice-Break Example ( (2 of 42 of 4))

units 1,826 = 0.02(1.20)

4)2(10,000)( =

iC

2DS = QOPT

Annual Demand (D)= 10,000 unitsCost to place an order (S)= $4

First, plug data into formula for each price-break value of “C”

units 2,000 = 0.02(1.00)

4)2(10,000)( =

iC

2DS = QOPT

units 2,020 = 0.02(0.98)

4)2(10,000)( =

iC

2DS = QOPT

Carrying cost % of total cost (i)= 2%Cost per unit (C) = $1.20, $1.00, $0.98

Interval from 0 to 2499, the Qopt value is feasible

Interval from 2500-3999, the Qopt value is not feasible

Interval from 4000 & more, the Qopt value is not feasible

Next, determine if the computed Qopt values are feasible or not

101

Price-Break ExamplePrice-Break Example 2 2 (3 (3 of 4 of 4))

Since the feasible solution occurred in the first price-break, it means that all the other true Qopt values occur at the beginnings of each price-break interval. Why?

Since the feasible solution occurred in the first price-break, it means that all the other true Qopt values occur at the beginnings of each price-break interval. Why?

0 1826 2500 4000 Order Quantity

Total annual costs

So the candidates for the price-breaks are 1826, 2500, and 4000 units

So the candidates for the price-breaks are 1826, 2500, and 4000 units

Because the total annual cost function is a “u” shaped function

Because the total annual cost function is a “u” shaped function

102

Price-Break Example Price-Break Example 22 ( (4 of 4 of 4)4)

iC 2

Q + S

Q

D + DC = TC iC

2

Q + S

Q

D + DC = TC

Next, we plug the true Qopt values into the total cost annual cost function to determine the total cost under each price-break

Next, we plug the true Qopt values into the total cost annual cost function to determine the total cost under each price-break

TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20) = $12,043.82TC(2500-3999)= $10,041TC(4000&more)= $9,949.20

TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20) = $12,043.82TC(2500-3999)= $10,041TC(4000&more)= $9,949.20

Finally, we select the least costly Qopt, which in this problem occurs in the 4000 & more interval. In summary, our optimal order quantity is 4000 units

Finally, we select the least costly Qopt, which in this problem occurs in the 4000 & more interval. In summary, our optimal order quantity is 4000 units

103

Multi-period Inventory Models:Multi-period Inventory Models:Fixed Time Period Fixed Time Period

(Fixed-Order- Interval) (Fixed-Order- Interval) ModelsModels

104

Orders are placed at fixed time intervals

Order quantity for next interval? (inventory is brought up to target amount, amount ordered varies)

Risk of stockout between intervals

Reasons for using the FOI model

– Supplier’s policy may encourage its use

– Grouping orders from the same supplier can produce savings in ordering, packing and shipping costs.

– Some circumstances do not lend themselves to continuously monitoring inventory position

– Requires only periodic checks of inventory levels (no continous monitoring is required)

Fixed-Order-Interval ModelFixed-Order-Interval Model

105

Inventory Level in a Fixed Period Inventory Level in a Fixed Period SystemSystem

Various amounts (Qi) are ordered at regular time intervals (p) based on the quantity necessary to bring inventory up to

target maximum

pp pp pp

QQ11 QQ22

QQ33

QQ44

Target maximum

TimeTime

d In

vent

ory

d In

vent

ory

106

Requires a larger safety stock Increases carrying cost Costs of periodic reviews

Fixed-Interval DisadvantagesFixed-Interval Disadvantages

107

Fixed-Quantity vs. Fixed-Interval Fixed-Quantity vs. Fixed-Interval Ordering

12-107

108

FOI ModelFOI Model

12-108

order of at timeposition Inventory

levelinventory Target

order Amount to

where

Order of Timeat PositionInventory

LevelInventoryTarget

Order toAmount

IP

T

Q

IPTQ

109

Fixed-Time Period Model with Safety Stock Fixed-Time Period Model with Safety Stock FormulaFormula

order)on items (includes levelinventory current = I

timelead and review over the demand ofdeviation standard =

yprobabilit service specified afor deviations standard ofnumber the= z

demanddaily averageforecast = d

daysin timelead = L

interval)(order ordersbetween timeoflength = T

ordered be toquantitiy = q

:Where

I - Z+ L)+(Td = q

L+T

L+T

order)on items (includes levelinventory current = I

timelead and review over the demand ofdeviation standard =

yprobabilit service specified afor deviations standard ofnumber the= z

demanddaily averageforecast = d

daysin timelead = L

interval)(order ordersbetween timeoflength = T

ordered be toquantitiy = q

:Where

I - Z+ L)+(Td = q

L+T

L+T

q = Average demand + Safety stock – Inventory currently on handq = Average demand + Safety stock – Inventory currently on hand

110

Fixed-Time Period Model: Fixed-Time Period Model: Determining the Value of Determining the Value of T+LT+L

T+L di 1

T+L

d

T+L d2

=

Since each day is independent and is constant,

= (T + L)

i

2

T+L di 1

T+L

d

T+L d2

=

Since each day is independent and is constant,

= (T + L)

i

2

The standard deviation of a sequence of random events equals the square root of the sum of the variances

111

Order Quantity for a Order Quantity for a Periodic Inventory SystemPeriodic Inventory System

Q = d(T + L) + zd T + L - Iwhere

d = average demand rateT = the fixed time between ordersL = lead time

d = standard deviation of demand

zd T + L = safety stockI = inventory level

z = the number of standard deviations for a specified service level

112

Fixed-Period Model with Variable Fixed-Period Model with Variable DemandDemand (Example 1 (Example 1))

d = 6 bottles per dayd = 1.2 bottlesT = 60 daysL = 5 daysI = 8 bottlesz = 1.65 (for a 95% service level)

Q = d(T + L) + zd T + L - I

= (6)(60 + 5) + (1.65)(1.2) 60 + 5 - 8

= 397.96 bottles

113

Fixed-Time Period ModelFixed-Time Period Model with withVariable Demand (Example 2)(1 of 3Variable Demand (Example 2)(1 of 3))

Average daily demand for a product is 20 units. The review period is 30 days, and lead time is 10 days. Management has set a policy of satisfying 96 percent of demand from items in stock. At the beginning of the review period there are 200 units in inventory. The standard deviation of daily demand is 4 units.

Given the information below, how many units should be ordered?

Given the information below, how many units should be ordered?

114

Fixed-Time Period ModeFixed-Time Period Model with Variable l with Variable Demand (Example 2)(2 of 3)Demand (Example 2)(2 of 3)

T+ L d2 2 = (T + L) = 30 + 10 4 = 25.298 T+ L d

2 2 = (T + L) = 30 + 10 4 = 25.298

So, by looking at the value from the Table, we have a probability of 0.9599, which is given by a z = 1.75

So, by looking at the value from the Table, we have a probability of 0.9599, which is given by a z = 1.75

115

Fixed-Time PeriodFixed-Time Period Model with Variable Model with Variable Demand (Example 2) Demand (Example 2) ((3 of 33 of 3))

or 644.272, = 200 - 44.272 800 = q

200- 298)(1.75)(25. + 10)+20(30 = q

I - Z+ L)+(Td = q L+T

units 645

or 644.272, = 200 - 44.272 800 = q

200- 298)(1.75)(25. + 10)+20(30 = q

I - Z+ L)+(Td = q L+T

units 645

So, to satisfy 96 percent of the demand, you should place an order of 645 units at this review period

So, to satisfy 96 percent of the demand, you should place an order of 645 units at this review period

116

Miscellaneous Systems:Miscellaneous Systems:Optional Replenishment SystemOptional Replenishment System

Maximum Inventory Level, M

MActual Inventory Level, I

q = M - I

I

Q = minimum acceptable order quantity

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

117

Single-Period Inventory ModelSingle-Period Inventory Model

118

• Single period model: model for ordering of perishables and other items with limited useful lives

• Shortage cost: generally the unrealized profits per unit

• Cshortage = Cs = Revenue per unit – Cost per unit

• Excess cost: difference between purchase cost and salvage value of items left over at the end of a period

• Cexcess = Ce = Cost per unit – Salvage value per unit

Single Period ModelSingle Period Model

119

Single-Period ModelSingle-Period Model

• The goal of the single-period model is to identify the order quantity that will minimize the long-run excess and shortage costs

• Two categories of problem:– Demand can be characterized by a continuous

distribution

– Demand can be characterized by a discrete distribution

12-119

120

• Continuous stocking levels– Identifies optimal stocking levels– Optimal stocking level balances unit shortage and

excess cost

• Discrete stocking levels– Service levels are discrete rather than continuous– Desired service level is equaled or exceeded

Single Period ModelSingle Period Model

121

Single-Period ModelSingle-Period Model

es

s

CC

CP

es

s

CC

CP

sold be unit will y that theProbabilit

estimatedunder demand ofunit per Cost C

estimatedover demand ofunit per Cost C

:Where

s

e

P

This model states that we should continue to increase the size of the inventory so long as the probability of selling the last unit added is equal to or greater than the ratio of: Cs/Cs+Ce

This model states that we should continue to increase the size of the inventory so long as the probability of selling the last unit added is equal to or greater than the ratio of: Cs/Cs+Ce

122

Optimal Stocking LevelOptimal Stocking Level

Service Level

So

Quantity

Ce Cs

Balance point

Service level =Cs

Cs + CeCs = Shortage cost per unitCe = Excess cost per unit

(Optimum Stocking Quantity)

123

Single Period Single Period Example Example 11• Ce = $0.20 per unit• Cs = $0.60 per unit• Service level = Cs/(Cs+Ce) = .6/(.6+.2)• Service level = .75

Service Level = 75%

Quantity

Ce Cs

Stockout risk = 1.00 – 0.75 = 0.25

124

Single Period Model ExampleSingle Period Model Example 2 2

Our college basketball team is playing in a tournament game this weekend. Based on our past experience we sell on average 2,400 shirts with a standard deviation of 350. We make $10 on every shirt we sell at the game, but lose $5 on every shirt not sold. How many shirts should we make for the game?

Cs = $10 and Ce = $5; P ≤ $10 / ($10 + $5) = .667

Z.667 = .432 therefore we need 2,400 + .432(350) = 2,551 shirts

125

Last WordsLast Words

Inventories have certain functions.

But too much inventory - Tends to hide problems- Costly to maintain

So it is desired• Reduce lot sizes• Reduce safety stocks

1 Inventory Management and Control. 2AMAZON.com Jeff Bezos, in 1995, started AMAZON.com as a...

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