Download - 04Inventory Management New

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Production Operations

Management

Fundamentals of Inventory

Management

From EOQ to ROP G. Ioannou, Ph.D.

Associate Professor


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Definition - Types

| Inventory is created when the receipt of materials, partsor finished goods exceeds their disbursement.

| The different types of inventory can be distinguished as:

z Raw Materials (RM)z Work-in-process (WIP)

z Finished goods (FG)

z Spare parts (SP)

| Each type has different attributes and, thus, requires adifferent management policy.


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Pressures for Low Inventories| The primary reason for keeping inventories low is that inventory

represents a temporary monetary investment in goods on which a firm

must pay (rather than receive) interest.| Inventory holding (or carrying) cost is the variable cost of keeping items

on hand, including interest, storage and handling, taxes, insurance andshrinkage.

| When these components change with inventory levels, so does theholding cost.

| Companies usually state an items holding cost per period of time as apercent of its value (it typically ranges from 20 to 40% of its value).

| Suppose that a firms holding cost is 30%. If the average value of total

inventory is 20% of sales, the average annual cost to hold inventory is6% (0.3x0.2) of total sales, which is sizable in terms of gross profitmargins, often less than 10%.

| Therefore, the components of holding cost create pressures for lowinventories.


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Pressures for Low Inventories| Interest or Opportunity Cost

z

To finance inventory, a company must obtain a loan or forgo theopportunity of an investment promising an attractive return.

z Interest or Opportunity Cost, whichever is greater, usually is thelargest component of holding cost, often as high as 15%.

|

Storage and Handling Costsz Inventory takes up space and must be moved into and out of

storage.

z Storage and handling costs may be incurred when a firm rentsspace on either a long- or short- term basis.

z There is also an opportunity cost for storage when a firm could usestorage space productively in some other way.


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Pressures for Low Inventories

| Taxes, Insurance and Shrinkage

z More taxes are paid if end-of-year inventories are high.

z Insurance on assets are increased when there is more to insure.

z Shrinkage takes three forms:

Pilferage: theft of inventory by customers or employees.

Obsolescence: inventory can not used or sold at full value becauseof model changes, engineering modifications or unexpectedly lowdemand.

Deterioration: physical spoilage or damage results in lost value.


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Pressures for High Inventories| Customer Service

z Creating inventory can speed delivery and improve on-timedelivery.

z Inventory reduces the potential for stock-outs and back-orders.

A Stockoutoccurs when an item that is typically stocked is not

available to satisfy a demand the moment it occurs, resulting inloss of sale.

A backorderis a customer order that can not be filled whenpromised or demanded but is filled later. Customers may bewilling to wait for a backorder but next time may take theirbusiness elsewhere. Sometimes customers are given discountsfor the inconvenience of waiting.


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Pressures for High Inventories

| Ordering Cost

z Each time a firm places a new order, it incurs an ordering cost,or the cost of preparing a purchase order for a supplier or aproduction order for the shop.

z For the same item, the ordering cost is the same, regardless of

the order size.z The purchasing agent needs to take the time to decide how

much to order, select a supplier and negotiate terms. Time isalso spent on paperwork, follow-up and receiving. In the case of

a production order for a manufactured item, a blueprint androuting instructions must often accompany the shop order.


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| Setup Cost

z The cost involved in changing over a machine to produce adifferent component or item is the setup cost. It includes labourand time to make the changeover, cleaning and new tools orfixtures.

z Scrap or rework costs can be substantially higher at the start ofthe run.

z Setup cost is also independent of order size.


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| Labor and Equipment Utilization

z By creating more inventory, management can increase workforceproductivity and facility utilization in three ways.

1. Placing larger, less frequent production orders reduces the numberof unproductive setups, which add no value to a product or service.

2. Holding inventory reduces the chance of costly rescheduling ofproduction orders because the components needed to make theproduct are not in inventory.

3. Building inventories improves resource utilization by stabilizing theoutput rate for industries when demand is cyclical or seasonal.

z The firm uses inventory built during slack periods to handle extrademand in peak seasons and minimizes the need for extra shifts,hiring, lay offs, overtime and additional equipment.


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Pressures for High Inventories| Transportation Cost

z

Transportation costs can be reduced by increasing inventory levels.z Having inventory on hand allows more carload shipments and minimizes

the need to expedite shipments by more expensive modes oftransportation.

z Combining orders of different items from same supplier may lead lead to

rate discounts, thereby decreasing the costs of transportation and rawmaterials.

| Payments to suppliers

z A firm often can reduce total payments to suppliers if it can tolerate higherinventory levels, if for example it learns that a key supplier is about toincrease prices.

z A firm can also take advantage of quantity discounts, whereby the price perunit drops when the order is sufficiently large.


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Economic Order Quantity

(EOQ)

| Managers face conflicting pressures to keep inventories low enough

to avoid excess inventory holding costs but high enough to reduce

the frequency of orders and setups.

| A good starting point for balancing these conflicting pressures and

determining the best cycle-inventory level for an item is finding theEconomic Order Quantity (EOQ).

| EOQ is defined as the lot size that minimizes total annual inventory

holding and ordering costs.


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EOQ Assumptions

| When the following assumptions hold, the optimal EOQ can be

calculated analytically:

z The model can only be used for "independent" demand and instantaneous

production.

z The demand rate is constant and known beforehand.

z There are no constraints in terms of lot size.

z The only relevant costs are the inventory holding cost and fixed cost per lot

for ordering or setup.

z There is no uncertainty in lead-time or supply (immediate).

| In reality, very few situations are so simple and well behaved.

| Still, EOQ is a reasonable first approximation of average lot sizes.


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Calculating EOQ

| When the EOQ assumptions are

satisfied, cycle inventory behavesas shown.

| A cycle begins with Q units held ininventory, which happens when a

new order is received.| During the cycle, on-hand inventory

is used at a constant rate.

| Since demand is known with

certainty and the lead time is aconstant, a new lot can be orderedso that inventory falls to 0 preciselywhen the new lot is received.

Q/2

Q

Receive

1

Q/2

Q

Order

1 cycle

ReceiveOrder

Inventory depletionDemand rate

Average

timeOn-handI

nventory

cycleinventory


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Calculating EOQ

| Because inventory varies

uniformly between Q and 0, theaverage cycle inventory equals

half the lot size, Q.

| The annual cost of holding

inventory increases linearly with

Q.

Annual holding cost = (Average cycle inventory) x(Unit holding cost)

Lot Size (Q)

AnnualCost,$

Holding Cost

HQ

AHC =

2


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Calculating EOQ

|The average number of ordersper year equals annual demand D

divided by Q.

| The total ordering or setup cost

decreases non-linearly as Qincreases, because fewer orders

are placed.

S

Q

DAOC =

Annual ordering cost = (Number of orders / year) x

(Ordering or Setup cost)

An

nualCost,$

Ordering Cost

Lot Size (Q)


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Calculating EOQ

Total cost = Annual holding cost + Annual ordering or setup cost

SQ

DH

QC +=

2

Annual

Cost

Lot size (Q)

Where:

C= total cost per year

Q = lot size in units

H= cost of holding one unit in

inventory for a year, often

calculated as a proportion of

the items value.

D = annual demand in units per year

S = cost of ordering or setting up one

lot, in dollars per lot.


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Calculating EOQ

SQ

D

H

Q

C += 2 0=dQ

dC

0)2(=

+

dQ

SQ

D

H

Q

d 022=+

HQSD

0

2)

2(

2

2

2

>=

+

=

H

SD

Q

H

Q

SDd

dQ

TCdH

DSEOQ

2=


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EOQ Reality

|

Everybody uses it| Why?

z It is simple

z It is robust (100% error in demand value, only 25%error in total cost!!!)

z It is embedded in all ERP and ex-MRP systems

z Holding and set-up costs are fairly insensitive to lot

size


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EOQ Example

| The Heat International Company produces air-conditioners.

| The air conditioning units it requires to this purpose are handled asindependent demand.

| The production can be assumed stable, as the company implements a Make-To-Stock policy (production continues normally during winter months).

| The demand rate for such air-conditioning units is 500 units per week.

| The air-conditioning units supplier charges 0.3 per unit, while the cost ofplacing a purchase order is 10.

| The annual cost of holding an air-conditioning unit in inventory is 25% of itsvalue.

| If the company orders 5,000 units when its inventory is exhausted, what is the

total cost?| How many air-conditioning units and how often should the company order to

minimize its total cost?


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EOQ Example

| The annual cost for ordering EOQ (2633 units) becomes:

| The time between successive orders (expressed in weeks) is:

48.197102633

26000075.0

2

2633

2=+=+= S

Q

DH

QC

weeksyearweeks

D

EOQTBOEOQ 26,552

26000

2633)/52( ==


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Quantity Discounts Model| Discounts offerings with quantity

constitute motives for ordering higher

quantities and, hence, holding largerinventories.

| The price does not remain constant,

as in the EOQ model, but varies along

with the ordering quantity.

| Therefore, it becomes crucial to find

the trade-off between price discount

and cost increase due to keeping

larger inventory.

| The total annual cost needs toincorporate except for the inventory

holding cost and ordering (or setup)

cost the purchase cost of the

materials studied.

| The total cost becomes:DPS

Q

DH

QC ++=

2

PD =$4

100 300 Q

C for = 4C 3.5

C 3

PD =$3.5 PD

=$3

200

EOQ 4EOQ 3.5EOQ 3

PD for=$4

100 300 Q

TotalCo

st

C 4C for = 3.5

C for = 3

PD for=$3.5 PD for

=$3

200

EOQ 4EOQ 3.5EOQ 3


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Quantity Discounts Model

| In order to determine the optimum lot size in the case of discounts

proportional to the quantity ordered, the following approach shouldbe followed:

z

Calculate EOQ, starting from the lowest price to the next higher price.z Check for feasibility (EOQ within the volume range from the respective

price).

z If the first feasible EOQ is found for the lowest price, it is optimal.

z Otherwise, calculate total cost for the first feasible EOQ and the largerprice break quantity at each lower price level.


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Quantity Discounts - Example| The retailer of Heat International is negotiating the air-conditioning

units respective purchase cost along with the quantity ordered.

| Heat International offers the discounts summarized in the tableprovided below:

| The total demand for air-conditioning units is 6,500 units.

| Ordering cost is 50 and the annual inventory holding cost is estimated25% of the units price.

| How many air-conditioning units should the retailer order to minimize itstotal annual cost?

Quantity Price /Unit

0 - 299 60

300-499 58,8500- 57


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Quantity Discounts - Example

| For the lowest price:

| For the next price:

| For the last price:

50057.2135725.0

5065002257


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Quantity Discounts - Example

| The total cost for EOQ is:

| The total cost when ordering 300 units:

| The total cost when ordering 500 units:

|

Since 374,712.5 is the minimum total annual cost, the optimum orderingquantity is 500 units.

5.122,39365006050208

65006025.0

2

208

2208 =++=++= PDS

Q

DH

QC

3.488,38565008.5850300

65008.5825.0

2

300300 =++=C

5.712,37465005750500

65005725.0

2

500500 =++=C


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Inventory Control Systems

| An Inventory Control System answers to the questions:

z How much should we order?z When should we order?

| In selecting an inventory control system for a particular application, thenature of the demands imposed on the inventory items is crucial.

| An important distinction between types of inventory is whether an item issubject to dependentorindependentdemand.

z Independentdemand is influence by external market conditions and is notrelated to the inventory decisions for any other item held in stock.

z Dependentdemand is required as components or inputs to a product or

service.

| Inventory control systems for independent demand inventory is the area wefocus in the following slides.


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Continuous Review System

(Q)

| A continuous review (Q) system orreorder point

(ROP) system or fixed order-quantity system tracks theremaining inventory of an item each time a withdrawal ismade to determine whether it is time to reorder.

| In practice, these reviews are done frequently (e.g. daily)and often continuously (after each withdrawal).

| At each review a decision is made about an itemsinventory position.

| If it is considered to be too low, the system triggers a neworder.


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(Q) definitions

| On-hand inventory OH = inventory count

| Schedule receipts SR = supplier orders expected

| Backorders BO = customer orders not yet filled

|

The Inventory Position (IP) measures the items abilityto satisfy future demand.

| It includes Scheduled Receipts (SR) plus On-Hand

inventory (OH) minus backorders (BO).

IP=OH+SR-BO


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(Q)

| When the inventory position reaches a predetermined

minimum level, called the Reorder point (R), a fixedquantity Q of the item is ordered.z Reorder point R = minimum allowed IP =

Average demand during supplier lead time

| In a continuous review system, although the order quantityQ is fixed, the time between orders can vary (it can bebased on the EOQ, a price break quantity, a container size,or some other quantity selected by management).

| Reorder point is a predetermined quantity of inventory.z It needs to be sufficient to cover the demand during the lead

time (the time interval between an order placement and anorder delivery).


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(Q)| The downward-sloping line

represents the on-hand inventory,

which is being the depleted at aconstant rate.

| When it reaches reorder point R(the horizontal line), a new order forQ units is placed.

| The on-hand inventory continues todrop throughout lead time L until theorder is received.

| At that time, which marks the end ofthe lead time, on-hand inventoryjumps by Q units.

| A new order arrives just wheninventory drops to 0.

| The time between orders (TBO) is

the same for each cycle.

TBO TBO TBO

R

L L L

TBO TBO TBO

R

L L L

orderdelivery

order

time

i

nventory

The Case of Deterministic Demand


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(Q)

| The inventory position IPcorresponds to the on-hand inventory, except

during the lead time.| Just after a new order is placed, at the start of the lead time, IPincreases

by Q, as shown by the dashed line.

| The IPexceeds OHby this same margin (Q) throughout the lead time.

| A possible exception is the situation when more than one scheduledreceipts is open at the same time, because of long lead times.

| At the end of the lead time, when the scheduled receipts convert to on-

hand inventory, IP=OHonce again.

| The key point here is to compare IP, not OH, with Rin deciding whether tore-order or not.

The Case of Deterministic Demand


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(Q)

| In reality, demand and lead times are not always predictable.

| This situation gives rise to the need for safety stocks.

| In this case, reorder point equals the average demand during lead time plus

the safety stock.

The Case of Stochastic Demand

R

L1 L2

2 3

L3time

inventory

R

order

delivery

order

1

L1 L22 3

L3

| The variability in lead times is assumed

negligible.| The wavy downward-sloping line

indicates that demand that varies from

day to day.

| Its slope is steeper in the second cycle,

which means that the demand rate is

higher during this time period.

| The changing demand rate means that

the time between order changes, so

TBO1 TBO2 TBO3


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(Q)| Because of uncertain demand, sales during lead time are unpredictable, and

safety stock is added to hedge against lost sales.

|

This addition is why Ris higher when demand is assumed stochastic incomparison with when demand is assumed deterministic.

| It also explains why the on-hand inventory usually drops to 0 by the time areplenishment order arrives.

| The greater the safety stock, and thus the higher reorder point R, the less

likely a stockout.| Because the average demand during lead time is variable and uncertain, the

real decision to be made when selecting Rconcerns the safety stock level.

| Deciding on a small or large safety stock is a trade-off between customerservice and inventory holding costs.

| Cost minimization models can be used to find the best safety stock, but theyrequire estimates of stock-out and backorders cost, which are usually difficultto make with any precision.

| The usual approach for determining Ris to set a reasonable service-levelpolicy for the inventory and then determine the safety stock level that satisfies

this policy.



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(Q)

| One way to determine the safety stock is to set a service level, or cycle service

level.| The service level is defined as the probability of not running out of stock in any

one ordering cycle, which begins at the time an order is placed and ends whenit arrives in stock.

| In a continuous review system the stockout risk that occurs only during lead

time is greater than the overall risk of stockout, because this risk is nonexistentoutside the ordering cycle.

| In order to translate this policy into a specific safety stock level, the demanddistribution must be known.

| Indeed, its variability plays a major role. It is measured with probabilitydistributions, specified by a mean and variance.

| When selecting the safety stock, the inventory planner often assumes thatdemand during lead time is normally distributed.



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(Q)

| Safety stock is computed by multiplying the number of standarddeviations from the mean needed to implement the cycle-service level zby the standard deviation of demand during lead time probability

distribution L:Safety stock= zxL

| The higher the value ofz, the higher the safety stock and the cycle-service level should be.

|

Ifz=0, there is no safety stock and stockouts will occur during 50% of theorder cycles.


R

zL

=85%

100-85=15%

R

average demandduringlead time

zL

servicelevel

=85% probabilityof not satisfying

demand=100-85=15%


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Q system Example (1)

| Heat International retailer has observed that the demand for air-

conditioning units is not exactly deterministic, but presents a level ofuncertainty.

| Following a relevant study, it has been found that the demand for air-

conditioning units during lead time has an average of 100 units and a

variance of 12.| How much units should Heat International keep in safety stock to

ensure that the service level is 99%?

| What should the reorder point be?


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1. Calculate z

zis the number of standard deviations to the right of average demand duringlead time that places 99% of the area under the curve to the left of that point

(0.9900 in the table of Normal Distribution).

The closest number in the table is 0.9901, which gives value of 2.33.

2. The safety stock becomes:Safety stock = zxL = 2.33x12=27.96 or 28 units.

3. The reorder point is calculated as following:

Reorder point = Average demand during lead time + Safety stock

R= 100+28 = 128 units


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(Q)| In case average demand during the lead time and the standard deviation of

demand during the lead time are not directly available:

| If

z The average demand dand the standard deviation of demand tover some

time interval tnot equal to the lead time L are known

z The probability distributions of demand for each time interval tare identical and

independent of each other.z The lead time L is expressed in terms of the time interval t

| Average demand during the lead time is the sum of the averages for each of

the L identical and independent distributions of demand, ord+d++d=dL

| The variance of the demand distribution for the lead time is the sum of thevariances of the L identical and independent distributions of demand, ort

2+

t2+...+ t

2= t2L

| The standard deviation of the demand distribution for the lead time is the

square root of the sum of their variances, or


LLttL

==2


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| Heat International retailer has observed that the demand for

air-conditioning units has an average of 125 units per weekand a standard deviation of 15 units per week.

| The lead time between an order placement and order deliveryis 4 weeks.

| The inventory holding cost of one air-conditioning unit for theretailer is 10, while the cost of placing an order is 50.

| How much inventory should the retailer keep in safety stock to

satisfy a service level of 90%?| How often should the retailer place an order and how many

air-conditioning units should he order?


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Q system Example (2)| The standard deviation of the demand distribution for the lead time is:

| From the tables of Normal Distribution for a service level of 90%, z= 1.28.| Safety stock, thus, derives as:

Safety stock = zx L = 1.28 x 30 = 38.4 or 39 units

| The reorder point becomes:

Re order point = Average demand during lead time + safety stockR= 125 x 4 + 39 = 539 units

| The annual demand is D = 125 units / week x 52 weeks = 6500 units

| Economic Order Quantity:

| Therefore, anytime the inventory drops to 539 units, the retailer needs to placean order for 255 units.

30415 === L tL

25510

05)251(5222

== H

DS

EOQ


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Periodic Review System (P)

| An alternative inventory control system is the Periodic review system

(P) orfixed interval reorder system orperiodic reorder system.| In the periodic review system (P) an items inventory position is

reviewed periodically rather than continuously.

| Such a system can simplify delivery scheduling, because it establishes

a routine.

| A new order is always placed at the end of each review and the time

between orders (TBO) is fixed at P.

| Demand is a random variable. So, demand between reviews varies.

| In a P system, the lot size Q may change from one order to the next,

but the time between orders is fixed.


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Periodic Review System (P)| From the original EOQ assumptions the following are maintained:

z There are no constraints on the size of the lot

z Decisions for one item are independent of decisions for other itemsz There is no uncertainty in lead times or supply.

delivery

L

11

Time

invento

ry

order

Order placed

L LP

Protection Interval

Q1

2

3

Q3Q2

P

| The downward-sloping line represents on-hand

inventory.

| When the predetermined time Phas elapsed sincethe last review, an order is placed to bring the

inventory position, represented by the dashed line, up

to the target level T.

| The lot size for the first review is:

Q1=T-IP1.| As with Q system, IPand OHdiffer only during lead

time.

| When the new order arrives, at the end of the lead

time, IPand OHare identical.

| Lot sizes vary from one cycle to the next.


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Periodic Review System (P)| To run a P system, managers must decide on the length of time between

successive reviews Pand the target inventory level T.

| The time between intervals Pcan be any convenient time (e.g. each Friday or everyother Friday) or TBOEOQ or TBO based on any other inventory system.

| An order must be large enough to make the inventory position IPlast beyond thenext review, which is Pperiods away.

| The checker must wait Pperiods to revise, correct and reestablish the inventory

position.| Then a new order is placed, but it does not arrive until after the lead time L.

| Therefore, the time interval for which inventory must be planned when each neworder is placed (protection interval) is P+L periods.

| This longer protection interval is a fundamental difference of P-system with Q-

system.| This longer protection interval additionally induces the need for higher safety stock.

| Therefore, the target inventory level Tderives as:

T= d(P+L) + (Safety Stock for protection interval) or

T= d(P+L) + zP+L

= d(P+L) + zt LP

+


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P System Example

| Heat International retailer sells a specific air-conditioning unit with a demand

rate of 125 units per week with a standard deviation of 15 units.| The lead time (time interval between order placement and order delivery) is 4

weeks.

| How much inventory should the retailer keep in safety stock to satisfy a 90%

service level?| How often should the retailer place a new order and how many air-conditioning

units should be ordered each time?

| What happens if Heat International retailer uses a periodic review system,

instead of the continuous review system initially assumed?

| Please compare the two systems.


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P System Example| The annual demand is: D = 125 units / week 52 weeks (per year) = 6500 units.

| The length of time between reviews is:

P= (EOQ/D)x52 weeks = 255 / 6500x52 = 2.04 2 weeks| The standard deviation for the protection interval P+L is:

|

For a service level of 90%, z= 1.28.| Therefore, the target inventory level Tbecomes:

T= d(P+L) + (Safety Stock for protection interval) =

= d(P+L) + zP+L

= (125x6) + 1.28x37 = 797.36 or 798 units

| In the case of P-system, the safety stock held is 47 (=1.28x37) units, which islarger than the safety stock held in the case of Q-system (39 units).

374215 +=+=+

LPLLP


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P-system advantages

1. Administration of the system is convenient, because

replenishments are made at fixed intervals.

2. Order for multiple items from the same supplier may be

combined into a single purchase order. This approach

reduces ordering and transportation costs and mayresult in a price break from the supplier.

3. The inventory position needs to be known only when a

review is made and not continuously.


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Q-system advantages

1. The review frequency of each item may be

individualized. Tailoring the review frequency to the itemcan reduce total ordering and holding costs.

2. Fixed lot sizes, if large enough, may result in quantity

discounts. Physical limitations, such as truckloadcapacities, materials handling methods and furnace

capacities also may require a fixed lot size.

3. Lower safety stocks result in savings.


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Hybrid Systems| Various hybrid inventory control systems merge some but not all the features

ofPand Q systems.

| Optional Replenishment system

A system used to review the inventory position at fixed time in intervals and, ifthe position has dropped to (or below) a predetermined level, to place avariable-sized order to cover expected needs. The new order is large enough tobring the inventory position up to a target inventory, similar to Tfor P-system.Still, orders are not placed after a review unless the inventory position has

dropped to the predetermined minimum level (acts as the reorder point Rin Q-system).

| Base-Stock system

A system that issues a replenishment orderQ each time a withdrawal is made,for the exact amount of the withdrawal. This one-for-one replacement policy

maintains the inventory position at a base-stock level equal to expecteddemand during lead time plus safety stock (acts as reorder point in Q system).However, order sizes now vary to keep the inventory position at Rat all times.Since this position is the lowest IPpossible that will maintain a specified servicelevel, the base-stock system may be used to minimize cycle inventory.