Inventory Management

Inventory Managemen

t

Learning ObjectivesLearning Objectives

Define the term inventory and list the major reasons for holding inventories; and list the main requirements for effective inventory management.

Discuss the nature and importance of service inventories

Discuss periodic and perpetual review systems. Discuss the objectives of inventory management. Describe the A-B-C approach and explain how it

is useful.

Learning ObjectivesLearning Objectives

Describe the basic EOQ model and its assumptions and solve typical problems.

Describe the economic production quantity model and solve typical problems.

Describe the quantity discount model and solve typical problems.

Describe reorder point models and solve typical problems.

Describe situations in which the single-period model would be appropriate, and solve typical problems.

Independent Demand

A

B(4) C(2)

D(2) E(1) D(3) F(2)

Dependent Demand

Independent demand is uncertain. Dependent demand is certain.

Inventory: a stock or store of goods

InventoryInventory

Inventory ModelsInventory Models

Independent demand – finished goods, items that are ready to be sold E.g. a computer

Dependent demand – components of finished products E.g. parts that make up the computer

Types of InventoriesTypes of Inventories

Raw materials & purchased parts

Partially completed goods called work in progress

Finished-goods inventories (manufacturing firms)

or merchandise (retail stores)

Types of Inventories (Cont’d)Types of Inventories (Cont’d)

Replacement parts, tools, & supplies

Goods-in-transit to warehouses or customers

Functions of InventoryFunctions of Inventory

To meet anticipated demand

To smooth production requirements

To decouple operations

To protect against stock-outs

Functions of Inventory (Cont’d)Functions of Inventory (Cont’d)

To take advantage of order cycles

To help hedge against price increases

To permit operations

To take advantage of quantity discounts

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

Inventory turnover is the ratio ofaverage annual cost of goods sold toaverage inventory investment.

A system to keep track of inventory

A reliable forecast of demand

Knowledge of lead times

Reasonable estimates of Holding costs

Ordering costs

Shortage costs

A classification system

Effective Inventory ManagementEffective Inventory Management

Inventory Counting SystemsInventory Counting Systems

Periodic SystemPhysical count of items made at periodic intervals

Perpetual Inventory System System that keeps track of removals from inventory continuously, thus monitoringcurrent levels of each item

Perpetual Inventory SystemPerpetual Inventory System

Two-Bin System - Two containers of inventory; reorder when the first is empty

Universal Bar Code - Bar code printed on a label that hasinformation about the item to which it is attached 0

214800 232087768

Lead time: time interval between ordering and receiving the order

Holding (carrying) costs: cost to carry an item in inventory for a length of time, usually a year

Ordering costs: costs of ordering and receiving inventory

Shortage costs: costs when demand exceeds supply

Key Inventory TermsKey Inventory Terms

ABC Classification SystemABC Classification SystemClassifying inventory according to some measure of importance and allocating control efforts accordingly.

AA – 10-20% items, 60-70% value

BB – 30-40% items, 20-30% value

CC – 50-60% items, 10-15% value Annual

Valueof items

AA

BB

CC

High

Low

Low HighPercentage of Items

Cycle CountingCycle Counting

A physical count of items in inventory

Cycle counting management

How much accuracy is needed?

When should cycle counting be performed?

Who should do it?

Cycle Counting ExampleCycle Counting Example5,000 items in inventory, 5,000 items in inventory, 500 A500 A items, items, 1,750 B1,750 B items, items, 2,750 C2,750 C items items

Policy is to count Policy is to count

A items every month (20 working days), A items every month (20 working days),

B items every quarter (60 days), and B items every quarter (60 days), and

C items every six months (120 days)C items every six months (120 days)Item

Class Quantity Cycle Counting PolicyNumber of Items Counted per Day

A 500 Each month 500/20 = 25/day

B 1,750 Each quarter 1,750/60 = 29/day

C 2,750 Every 6 months 2,750/120 = 23/day

77/day

Economic order quantity (EOQ) model

The order size that minimizes total annual cost

Economic production model

Quantity discount model

Economic Order Quantity ModelsEconomic Order Quantity Models

Only one product is involved

Annual demand requirements known

Demand is even throughout the year

Lead time does not vary

Each order is received in a single delivery

There are no quantity discounts

Assumptions of EOQ ModelAssumptions of EOQ Model

The Inventory CycleThe Inventory Cycle

Profile of Inventory Level Over Time

Quantityon hand

Q

Receive order

Placeorder

Receive order

Placeorder

Receive order

Lead time

Reorderpoint

Usage rate

Time

Total CostTotal Cost

Annualcarryingcost

Annualorderingcost

Total cost = +

TC = Q2

H DQ

S+

Q = Order quantityH = Holding costD = Annual demandS = Ordering cost or Setup cost

Cost Minimization GoalCost Minimization Goal

Order Quantity (Q)

The Total-Cost Curve is U-Shaped

Ordering Costs

QO

An

nu

al C

os

t

(optimal order quantity)

TCQH

D

QS

2

Deriving the EOQDeriving the EOQ

Using calculus, we take the derivative of the total cost function and set the derivative (slope) equal to zero and solve for Q.

Q = 2DS

H =

2(Annual Demand)(Order or Setup Cost)

Annual Holding CostOPT

Minimum Total CostMinimum Total Cost

The total cost curve reaches its minimum where the carrying and ordering costs are equal.

Q2

H DQ

S=

Reorder PointsReorder Points

EOQEOQ answers the “ answers the “how muchhow much” question” question

The reorder point (The reorder point (ROPROP) tells ) tells whenwhen to to orderorder

ROP ROP ==Lead time for a Lead time for a

new order in daysnew order in daysDemand Demand per dayper day

== d x L d x L

d = d = DDNumber of working days in a yearNumber of working days in a year

Reorder Point CurveReorder Point Curve

Q*Q*

ROP ROP (units)(units)In

ven

tory

lev

el (

un

its)

Inve

nto

ry l

evel

(u

nit

s)

Time (days)Time (days)Lead time = LLead time = L

Slope = units/day = dSlope = units/day = d

Production done in batches or lots Capacity to produce a part exceeds the

part’s usage or demand rate Assumptions of EPQ are similar to EOQ

except orders are received incrementally during production

Economic Production Quantity (EPQ)Economic Production Quantity (EPQ)

Only one item is involved Annual demand is known Usage rate is constant Usage occurs continually Production rate is constant Lead time does not vary No quantity discounts

Economic Production Quantity Economic Production Quantity AssumptionsAssumptions

Production Order Quantity Production Order Quantity ModelModel

Inve

nto

ry le

vel

Inve

nto

ry le

vel

TimeTime

Demand part of Demand part of cycle with no cycle with no productionproduction

Part of inventory cycle Part of inventory cycle during which production during which production (and usage) is taking place(and usage) is taking place

tt

Maximum Maximum inventoryinventory

Economic Run SizeEconomic Run Size

QDS

H

p

p u0

2

Q = Order quantityH = Holding costD = Annual demandS = Ordering cost or Setup costp = Production rate or Delivery rateu = Usage rate

Quantity discount modelQuantity discount model

Annualcarryingcost

PurchasingcostTC = +

Q2

H DQ

STC = +

+Annualorderingcost

PD +

Quantity discount modelQuantity discount model

Unit price 1Unit price 2

Unit price 3

TC1

TC3

TC2

Cost

Quantity

When to Reorder with EOQ When to Reorder with EOQ OrderingOrdering

Reorder Point - When the quantity on hand of an item drops to this amount, the item is reordered

Safety Stock - Stock that is held in excess of expected demand due to variable demand rate and/or lead time.

Service Level - Probability that demand will not exceed supply during lead time.

Determinants of the Reorder Determinants of the Reorder PointPoint

The rate of demand The lead time Demand and/or lead time variability Stockout risk (safety stock)

Stock-out Cost can be Stock-out Cost can be determineddetermined

Used when demand is not constant or Used when demand is not constant or certaincertain

Use safety stock to achieve a desired Use safety stock to achieve a desired service level and avoid stockoutsservice level and avoid stockouts

ROP ROP == d x L d x L + + ssss

Annual stockout costs = the sum of the Annual stockout costs = the sum of the {{units shortunits short x x the probabilitythe probability

x x the stockout cost/unitthe stockout cost/unit x x the number of orders per yearthe number of orders per year}}

Safety Stock ExampleSafety Stock Example

Demand (Units)Probability of demand level

30 .2

40 .2

ROP 50 .3

60 .2

70 .1

1.0

ROP ROP = 50= 50 units units Stockout cost Stockout cost = $40= $40 per units per unitsOrders per year Orders per year = 6= 6 Carrying cost Carrying cost = $5= $5 per units per year per units per year

Safety Stock ExampleSafety Stock Example

ROP ROP = 50= 50 units units Stockout cost Stockout cost = $40= $40 per frame per frameOrders per year Orders per year = 6= 6 Carrying cost Carrying cost = $5= $5 per frame per year per frame per year

Safety Stock

Additional Holding Cost Stockout Cost

Total Cost

20 (20)($5) = $100 $0 $100

10 (10)($5) = $50 (10)(.1)($40)(6) = $240 $290

0 $0 (10)(.2)($40)(6) + (20)(.1)($40)(6) = $960 $960

A safety stock of A safety stock of 2020 frames gives the lowest total cost frames gives the lowest total cost

ROP ROP = 50 + 20 = 70= 50 + 20 = 70 frames frames

Safety stock

ROP ROP

Place Place orderorder

Probabilistic DemandProbabilistic DemandIn

ven

tory

lev

elIn

ven

tory

lev

el

TimeTime

Minimum demand during lead timeMinimum demand during lead time

Maximum demand during lead timeMaximum demand during lead time

Mean demand during lead timeMean demand during lead time

Normal distribution probability of Normal distribution probability of demand during lead timedemand during lead time

Expected Expected demand demand during during lead timelead time

ROPROP

Receive Receive orderorder

Lead Lead timetime

00

Risk

Probabilistic DemandProbabilistic Demand

Safety Safety stockstock

Probability ofProbability ofno stockoutno stockout

95% of the time95% of the time

Mean Mean demanddemand

ROP = ? unitsROP = ? units QuantityQuantity

Number of Number of standard deviationsstandard deviations

00 zz

Risk of a stockout Risk of a stockout (5% of area of (5% of area of normal curve)normal curve)

Z=1.65

Probabilistic DemandProbabilistic Demand

Use prescribed service levels to set safety Use prescribed service levels to set safety stock when the stock when the cost of stockouts cannot be cost of stockouts cannot be determineddetermined

ROP = demand during lead time + ZROP = demand during lead time + Zdltdlt

wherewhere Z Z ==number of standard number of standard deviationsdeviations

dltdlt = =standard deviation of standard deviation of demand during lead timedemand during lead time

Other Probabilistic ModelsOther Probabilistic Models

1.1. When demand is variable and lead When demand is variable and lead time is constanttime is constant

2.2. When lead time is variable and When lead time is variable and demand is constantdemand is constant

3.3. When both demand and lead time When both demand and lead time are variableare variable

When data on demand during lead time is When data on demand during lead time is not available, there are other models not available, there are other models availableavailable


Demand is variable and lead time is constantDemand is variable and lead time is constant

ROP ROP == ((average daily demand average daily demand x lead time in daysx lead time in days) +) + Z Zdltdlt

wherewhere dd == standard deviation of demand per day standard deviation of demand per day

dltdlt = = dd lead timelead time

Probabilistic ExampleProbabilistic Example

Average daily demand Average daily demand ((normally distributednormally distributed) = 15) = 15Standard deviation Standard deviation = 5= 5Lead time is constant at Lead time is constant at 22 days days90%90% service level desired service level desired

Z for Z for 90%90% = 1.28= 1.28

ROPROP = (15 = (15 units x units x 22 days days) +) + Z Zdltdlt

= 30 + 1.28(5)( 2)= 30 + 1.28(5)( 2)

= 30 + 8.96 = 38.96 ≈ 39= 30 + 8.96 = 38.96 ≈ 39

Safety stock is about Safety stock is about 99 units units


Lead time is variable and demand is constantLead time is variable and demand is constant

wherewhere ltlt == standard deviation of lead time in days standard deviation of lead time in days

ROP ROP == ((daily demand x average lead timedaily demand x average lead time) ) ++ Z Zdltdlt

ZZdlt dlt = Z xZ x ( (daily demanddaily demand) ) xx σσltlt


Daily demand Daily demand ((constantconstant) = 10) = 10Average lead time Average lead time = 6= 6 days daysStandard deviation of lead time Standard deviation of lead time = = ltlt = 3 = 398%98% service level desired service level desired

Z for Z for 98%98% = 2.055= 2.055

ROPROP = (10 = (10 units x units x 66 days days) + 2.055(10) + 2.055(10 units units)(3))(3)

= 60 + 61.55 = 121.65= 60 + 61.55 = 121.65

Reorder point is about Reorder point is about 122 122 unitsunits


Both demand and lead time are variableBoth demand and lead time are variable

ROP ROP == ((average daily demand average daily demand x average lead timex average lead time) +) + Z Zdltdlt

wherewhere dd == standard deviation of demand per daystandard deviation of demand per day

ltlt == standard deviation of lead time in daysstandard deviation of lead time in days

dltdlt == ((average lead time x average lead time x dd22) )

+ (+ (average daily demandaverage daily demand)) 2 2ltlt22


Average daily demand Average daily demand ((normally distributednormally distributed) = 150) = 150Standard deviation Standard deviation = = dd = 16 = 16Average lead time Average lead time 55 days days ((normally distributednormally distributed))Standard deviationStandard deviation = = ltlt = 1 = 1 daysdays95%95% service level desired service level desired

Z for Z for 95%95% = 1.65= 1.65

ROPROP = (150 = (150 packs x 5 dayspacks x 5 days) + 1.65) + 1.65dltdlt

= (150 x 5) + 1.65 (5 days x 16= (150 x 5) + 1.65 (5 days x 1622) + (150) + (15022 x 1 x 122))

= 750 + 1.65(154) = 1,004 = 750 + 1.65(154) = 1,004 packspacks

Orders are placed at fixed time intervals Order quantity for next interval? Suppliers might encourage fixed

intervals May require only periodic checks of

inventory levels Risk of stockout

Fixed-Order-Interval ModelFixed-Order-Interval Model

On

-han

d i

nve

nto

ryO

n-h

and

in

ven

tory

TimeTime

QQ11

QQ22

Target maximum Target maximum ((TT))

PP

QQ33

QQ44

PP

PP

Fixed-Order-Interval ModelFixed-Order-Interval Model

Tight control of inventory items Items from same supplier may yield

savings in: Ordering Packing Shipping costs

May be practical when inventories cannot be closely monitored

Fixed-Interval BenefitsFixed-Interval Benefits

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

Fixed-Interval DisadvantagesFixed-Interval Disadvantages

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

Shortage cost: generally the unrealized profits per unit

= 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

= Original cost per unit – Salvage value per unit

Single Period ModelSingle Period Model

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

Example 15Example 15

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

SupermarketSupermarket

Items used by many products are Items used by many products are held in a common area often called a held in a common area often called a supermarketsupermarket

Items are withdrawn as neededItems are withdrawn as needed

Inventory is maintained using JIT Inventory is maintained using JIT systems and proceduressystems and procedures

Common items are not planned by Common items are not planned by the MRP systemthe MRP system

Too much inventory Tends to hide problems Easier to live with problems than to

eliminate them Costly to maintain

Wise strategy Reduce lot sizes Reduce safety stock

Operations StrategyOperations Strategy

Inventory Management

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