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Chapter 17. Inventory Control

InventoryInventory is the stock of any item or resource used in an organization and can include: raw materials, finished products, component parts, supplies, and work-in-process

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

Firms invest 25-35 percent of assets in inventory but many do not manage inventories well

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Purposes of Inventory

1. To maintain independence of operations Provide “optimal” amount of cushion between work

centers Ensure smooth work flow

2. To allow flexibility in production scheduling 3. To meet variation in product demand 4. To provide a safeguard for variation in raw

material or parts delivery time Protect against supply delivery problems (strikes,

weather, natural disasters, war, etc.)

5. To take advantage of economic purchase-order size

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Independent vs. Dependent Demand

Inventory costs

Single-Period Model

Multi-Period Models: Basic Fixed-Order Quantity Models

Event triggered (Example: running out of stock, or dropping

below a “reorder point”)

EOQ, EOQ with reorder point (ROP) , and with safety stock

Multi-Period Models: Basic Fixed-Time Period Model

EOQ with Quantity Discounts

ABC analysis

Inventory Control (Management)

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E(1)

Independent vs. Dependent Demand

Independent Demand (Demand not related to other items or the final end-product)

Dependent Demand(Derived demand

items for component parts,

subassemblies, raw materials, etc.)

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Inventory Costs

Holding (or carrying) costs. Costs for capital, taxes, insurance, etc. (Dealing with storage and handling)

Setup (or production change) costs. (manufacturing) Costs for arranging specific equipment setups, etc.

Ordering costs (services & manufacturing) Costs of someone placing an order, etc.

Shortage (backordering) costs. Costs of canceling an order, customer goodwill, etc.

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A Single-Period Model Sometimes referred to as the newsboy problem Is used to handle ordering of perishables (fresh seafood,

cut flowers, etc.) and items that have a limited useful life (newspaper, magazines, high fashion goods, some high tech components, etc)

The optimal stocking level uses marginal analysis is where the expected profit (benefit from derived from carrying the next unit) is less than the expected cost of that unit (minus salvage value)

Co = Cost/unit of overestimated demand (excess demand)Co = Cost per unit – salvage value per unit

Cu = Cost/unit of underestimated demand Cu = Price/unit – cost/unit + cost of loss of goodwill per

unit Optimal order level is where P <= Cu /(Co + Cu )

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: Cu/Co+Cu

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Single Period Model Example

UNC Charlotte 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?

1. Determine Cu = $10 and Co = $5 (this time, these were

directly given)

2. Compute P ≤ $10 / ($10 + $5) = 0.667 66.7%3. Order up to ~ 66.7% of the demand4. How do you determine it? 5. Normal distribution, Z transformation,6. Z0.667 = 0.432 (use NORMSDIST(.667) or Appendix E)

7. Therefore we need 2,400 +0.432(350) = 2,551 shirts

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Single Period Model, Marginal Analysis Marginal analysis approach.

Consider solved problem 1, p. 6171. Determine Cu = 100-70 = $30 and Co = 70-20 = $502. Compute P ≤ 30/(30+50) 0.3753. Develop a full marginal analysis table (Excel time!)4. Assume we purchase 35 units, compute the expected total cost

5. Repeat step 4, for 36,…, 40

The optimal order (purchase) size is the no. of units with the minimum expected total cost

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Fixed-Order Quantity Models: Assumptions

Demand for the product is constant and uniform throughout the period.

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.)

Lead time (time from ordering to receipt) is constant (later, this assumption is relaxed with “safety stocks”).

Price per unit of product is constant.

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Basic Fixed-Order Quantity Model and Reorder Point Behavior

R = Reorder pointQ = Economic order quantityL = Lead time

L L

Q QQ

R

Time

Numberof unitson hand

1. You receive an order quantity Q.

2. Your start using them up over time. 3. When you reach down to

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

4. The cycle then repeats.

17-10

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Cost Minimization Goal

Ordering Costs

HoldingCosts

QOPTIMAL Order Quantity (Q)

COST

Annual Cost ofItems (DC)

Total Cost

By adding the item, holding, and ordering costs together, we determine the total cost curve, which in turn is used to find the Qoptimal (a.k.a. “EOQ”) inventory order point that minimizes total costs.

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Basic Fixed-Order Quantity (EOQ) Model

AnnualHolding

CostTotal Annual Cost =

AnnualPurchase

Cost

AnnualOrdering

Cost+ +

SQ

DH

QDCTC

2

A little bit of calculus…

H

DSEOQ

2

Ld=ROP _

A little bit of common sense…

LzLd=ROP _

ROP with safety stock…

TC = Total annual costD = DemandC = Cost per unitQ = Order quantityS = Cost of placing an order or setup costH = Annual holding and storage cost per unit

of inventory R or ROP = Reorder pointL = Lead time (constant) = average (daily, weekly, etc) demand

σL = Standard deviation of demand during lead time

_

d

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Basic EOQ & ROP Example

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, reorder point, and total annual cost?

EOQ 89.44 89 or 90 unitsROP 2.74*7 19.18 19 or 20 units

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Another example

Days per year considered in average daily demand = 360Average daily demand is 3.5 unitsStandard deviation of daily demand is 0.95 unitsCost to place an order = $50Holding cost per unit per year = $7.25Lead time = 4 days

Compute the EOQ, and ROP is the firm wants to maintain a 97% service levelservice level (probability of not stocking out)

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d

1

2

constant, is andt independen isday each Since

dL

L

idL

L

i

LzLd=ROP _

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Fixed-Time Period Model with Safety Stock

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 toquantitiy = q

:Where

I - Z+ L)+(Td = q

L+T

L+T

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

2dL+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 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.

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

25.298 =410+30 =L)+(T = 22dL+T

q = 20(30+10) + 1.75(25.30) – 200 644.27 units

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A special purpose model

Price-Break Model (Quantity discounts) Based on the same assumptions as the EOQ model, the

price-break model has a similar EOQ (Qopt) formula:

Annual holding cost, H, is calculated using H = iC where i = percentage of unit cost attributed to carrying inventory C = cost per unit

Since “C” changes for each price-break, the formula above must be applied to each price-break cost value.

Determine the total cost for each price break The lowest total cost suggests the optimal order size

(EOQ)

Cost Holding Annual

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

iC

2DS = QOPT

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Price-Break Example

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 $0.98

Re-do the example with an order cost of $25 and an inventory carrying cost rate of 45%.

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0 1826 2500 4000 Order Quantity

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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

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 65% are the “C” items

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30

60

30

60

AB

C

% of $ Value

% of Use

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Inventory Accuracy and Cycle Counting

Inventory accuracy refers to how well the inventory records agree with physical count Lock the storeroom Hire the right personnel for as storeroom manager or

employees Cycle Counting is a physical inventory-taking

technique in which inventory is counted on a frequent basis rather than 1-2 times a year Easier to conduct when inventories are low Randomly (minimize predictability) Pay more attention to A items, then B, etc.

Suggested problems: 3, 6, 12, 14, 17, 18, 21, 24

Case: Hewlett-Packard