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What is Inventory?

Definition--The stock of any item or resource used in an organization Raw materials

Finished products

Component parts

Supplies

Work in process

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Inventory System Purpose

The set of policies and controls that determine what inventory levels should be maintained, when stock should be replenished, and how large orders should be

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Purposes of Inventory

1. To maintain independence of operations

2. To meet variation in product demand

3. To allow flexibility in production scheduling

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

5. To take advantage of economic purchase-order size

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Inventory Costs Holding (or carrying) costs

Setup (or production change) costs

Ordering costs

Shortage (or backlog) costs

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Independent vs. Dependent Demand

Independent Demand(Demand not related to other items)

Dependent Demand(Derived/Calculated)

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Classifying Inventory Models

Fixed-Order Quantity Models Event triggered Make exactly the same amount Use re-order point to determine

timing

Fixed-Time Period Models Time triggered Count the number needed to re-order

Inventory Control

Inventory Inventory Models

Fixed Time Period Models

Fixed Order Quantity Models

Uncertainty in Demand

Find the EOQ and R

SimpleEOQ

EOQw/usage

EOQ w/Quantity

Discounts

Select Q and find R

Find the EOQ and R

Determinep and d

CalculateTotal costs

Constant Demand

Single Period Models

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Fixed-Order Quantity ModelsAssumptions

Demand for the product is constant and uniform throughout the period

Lead time (time from ordering to receipt) is constant

Price per unit of product is constant Inventory holding cost is based on average

inventory Ordering or setup costs are constant All demands for the product will be satisfied

(No back orders are allowed)

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EOQ Model--BasicFixed-Order Quantity Model

R = Reorder pointQ = Economic order quantityL = Lead time

L L

Q QQ

R

Time

InventoryLevel

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Basic Fixed-Order Quantity Model

Total Annual Cost =

AnnualPurchase

Cost

AnnualOrdering

Cost

AnnualHolding

Cost+ +

Derive the Total annual Cost Equation, where:TC - Total annual costD - Annual demand (and d-bar = average daily demand = D/365)C - 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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Cost Minimization Goal

Ordering Costs

HoldingCosts

QOPT

Order Quantity (Q)

COST

Annual Cost ofItems (DC)

Total Cost

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Deriving the EOQ Using calculus, we take the derivative of the total cost

function and set the derivative (slope) equal to zero

L = Lead time (constant)

d = average demand per time unit _

Cost Holding Annual

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

H

2DS = QOPT

Reorder Point, R = dL

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EOQ Example

Annual Demand (D) = 1,000 unitsDays per year considered in average daily demand = 365Cost to place an order (S) = $10Holding cost per unit per year (H) = $2.40Lead time (L) = 7 daysCost per unit (C) = $15

Determine the economic order quantity and the reorder point.

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Solution

units 91.287 = 2.40

)(10) 2(1,000 =

H

2DS = QOPT

d = 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

When the inventory level reaches 20, order 91 units.

91 or 92 units???

Why do we round up?

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Problem

Retailer of Satellite DishesD = 1000 unitsS = $ 25H = $ 100

How much should we order?

What are the Total Annual Stocking Costs?

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EOQ with Quantity Discounts

What if we get a price break for buying a larger quantity?

To find the lowest cost order quantity: Since “C” changes for each price-break, H=iC Where, i = percentage of unit cost attributed to

carrying inventory and , C = cost (or price) per unit Find the EOQ at each price break. Identify relevant and feasible order quantities. Compare total annual costs The lowest cost wins.

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EOQ with Quantity Discounts Example

Copper may be purchased for $ .82 per pound for up to 2,499 pounds$ .81 per pound for 2,500 to 5,000 pounds$ .80 per pound for orders greater than 5,000

poundsDemand (D) = 50,000 pounds per yearHolding costs (H) are 20% of the purchase price per

unitOrdering costs (S) = $30

How much should the company order to minimize total costs?

Problem 28

40

41

42

43

44

0 20 40 60 80 100

(Order Quantity 100's of units)

(Co

sts

in $

,000

)

<2500

<2500 - 4999

>5000

Feasible

Inventory Control

Inventory Inventory Models

Fixed Time Period Models

Fixed Order Quantity Models

Uncertainty in Demand

Find theEOQ and R

Find the L

Find Z

Safety Stock

Constant Demand

Single Period Models

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What if demand is not Certain?

Use safety stock to cover uncertainty in demand. Given: service probability which is the probability

demand will NOT exceed some amount. The safety stock level is set by increasing the

reorder point by the amount of safety stock. The safety stock equals z•L

where,L = the standard deviation of demand during the

lead time.

For example for a 5% chance of running out z 1.65

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ProblemAnnual Demand = 25,750 or 515/wk @ 50

wks/yearAnnual Holding costs = 33% of item cost

($10/unit)Ordering costs are $250.00d = 25 per week Leadtime = 1 weekService Probability = 95%

Find:a.) the EOQ and Rb.) annual holding costs and annual setup costsc.) Would you accept a price break of $50 per

order for lot sizes that are larger than 2000?

Inventory Control

Inventory Inventory Models

Fixed Time Period Models

Fixed Order Quantity Models

Current Inventory

Find theT+L

Find Z

Find order quantity (q)

Single Period Models

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Fixed-Time Period Models Check the inventory every review period and

then order a quantity that is large enough to cover demand until the next order will come in.

The model assumes uncertainty in demand with safety stock added to the order quantity.

More exposure to variability than fixed-order models

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Fixed-Time Period Model with Safety Stock Formula

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

reviewsbetween days ofnumber the= T

ordered be oquantity t = q

:Where

I - Z+ L)+(Td = q

L+T

L+T

q = Average demand + Safety stock - Inventory currently on hand

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Determining the Value of T+L

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

2

dL+T

d

L+T

1i

2dL+T

L)+(T =

constant, is andt independen isday each Since

= i

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Example of the Fixed-Time Period Model

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 percentof 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.

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

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Example of the Fixed-Time Period Model: Solution

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.

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

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Problem A pharmacy orders antibiotics every two

weeks (14 days). the daily demand equals 2000 the daily standard deviation of demand =

800 lead time is 5 days service level is 99 % present inventory level is 25,000 units

What is the correct quantity to order to minimize costs?

Inventory Control

Inventory Inventory Models

Fixed Time Period Models

Fixed Order Quantity Models

Uncertainty in Demand

Constant Demand

Single Period Models

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Single – Period Model for items w/obsolescence (newsboy problem)

For a single purchase Amount to order is when marginal profit

(MP) is equal to marginal loss (ML). Adding probabilities (P = probability of that unit being sold) for the last unit ordered we want

P(MP)(1-P)ML or P ML /(MP+ML)

Increase order quantity as long as this holds.

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Single–Period Model (Text Prob. #21)

Famous Albert’s Cookie King

Demand(dozen)

Probability of Demand

1,800 0.05

2,000 0.10

2,200 0.20

2,400 0.30

2,600 0.20

2,800 0.10

3,000 0.05

How many cookiesshould he bake?

Each dozen sells for $0.69and costs $0.49 with a salvage value of $0.29.

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ABC Classification System

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

dollars invested

profit potential

sales or usage volume

stock-out penalties

0

30

60

30

60

AB

C

% of $ Value

% of Use

So, identify inventory items based on percentage of total dollar value, where “A” items are roughly top 15 %, “B” items as next 35 %, and the lower 50% are the “C” items.

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Inventory Accuracy and Cycle Counting

Inventory accuracy Do inventory records agree with

physical count?

Cycle Counting Frequent counts When? (zero balance, backorder,

specified level of activity, level of important item, etc.)