Inventory Inventory ManagementManagement

InventoryInventory management management

A subsystem of logisticsA subsystem of logistics Inventory: a stock of materials or Inventory: a stock of materials or

other goods tother goods too facilitate production facilitate production or to satisfy customer demandor to satisfy customer demand

Main decisions:Main decisions: Which items should be carried in stock?Which items should be carried in stock? How much should be ordered?How much should be ordered? When should an order be placed?When should an order be placed?

The need to hold stocks The need to hold stocks 11 BufferBuffer between Supply and Demand between Supply and Demand

To keep down production costsTo keep down production costs: to achieve low unit : to achieve low unit costs, production have to run as long as possible costs, production have to run as long as possible (setting up machines is tend to be costly)(setting up machines is tend to be costly)

To take account of variable supply times: safety To take account of variable supply times: safety stock to cover delivery delays from suppliersstock to cover delivery delays from suppliers

To minimize buying costs associated with raising an To minimize buying costs associated with raising an orderorder

To accommodate variations (on the short run) in To accommodate variations (on the short run) in demand (to avoid stock-outs)demand (to avoid stock-outs)

To account for seasonal fluctuations:To account for seasonal fluctuations: There are products popular only in peak timesThere are products popular only in peak times There are goods produced only at a certain time of the yearThere are goods produced only at a certain time of the year

Adaptation Adaptation ffo the o the fluctuation of demand with fluctuation of demand with

building up stocksbuilding up stocksDEMAND

CAPACITY

Inventory accumulation

Inventory reduction

The need to hold stocks The need to hold stocks 22

To take advantage of quantity discounts (buying in To take advantage of quantity discounts (buying in bulk)bulk)

To allow for price fluctuations/speculation: to buy To allow for price fluctuations/speculation: to buy large quantities when a good is cheaperlarge quantities when a good is cheaper

To help production and distribution operations run To help production and distribution operations run smoothly: to increase the independence of these smoothly: to increase the independence of these activitiesactivities

To provide immediate service for customersTo provide immediate service for customers To minimize production delays caused by lack of To minimize production delays caused by lack of

spare parts (for maintenance, breakdowns)spare parts (for maintenance, breakdowns) Work-in-progress: facilitating production process by Work-in-progress: facilitating production process by

providing semi-finished stocks between different providing semi-finished stocks between different processesprocesses

Types of Types of Stock-holding/InventoryStock-holding/Inventory

raw material, component and packaging stockraw material, component and packaging stock in-process stocks (work-in-progress; WIP)in-process stocks (work-in-progress; WIP) finished products (finished goods inventory; finished products (finished goods inventory;

FGI)FGI) pipeline stocks: held in the distribution chainpipeline stocks: held in the distribution chain general stores: contains a mixture of products general stores: contains a mixture of products

to support to support spare parts: spare parts:

Consumables (nuts, bolts, etc.)Consumables (nuts, bolts, etc.) Rotables and repairablesRotables and repairables

Independent vs. Independent vs. dependent dependent demanddemand

Independent demand:Independent demand: Influenced only by market conditionsInfluenced only by market conditions Independent from operationsIndependent from operations Example: finished goodsExample: finished goods

Dependent demand:Dependent demand: RelRelaated to the demand for another item ted to the demand for another item Example: product components, raw Example: product components, raw

materialmaterial

Another typology of Another typology of stocksstocks

working stockworking stock: reflects the actual : reflects the actual demanddemand

cycle stockcycle stock: follows the production (or : follows the production (or demand) cyclesdemand) cycles

safety stocksafety stock: to cover unexpected : to cover unexpected fluctuations in demandfluctuations in demand

speculative stockspeculative stock: built up on : built up on expectationsexpectations

seasonal stockseasonal stock: goods stockpiled before : goods stockpiled before peakspeaks

Inventory costInventory cost Item costItem cost: the cost of buying or producing : the cost of buying or producing

inventory itemsinventory items Ordering costOrdering cost: does not depend on the number of : does not depend on the number of

items ordered. Form typing the order to items ordered. Form typing the order to transportation and receiving costs.transportation and receiving costs.

Holding (carrying) costHolding (carrying) cost:: Capital cost: the opportunity cost of tying up Capital cost: the opportunity cost of tying up

capitalcapital Storage cost: space, insurance, taxStorage cost: space, insurance, tax Cost of obsolescence, deterioration and lossCost of obsolescence, deterioration and loss Sometimes designated by management rather Sometimes designated by management rather

than computedthan computed Stockout costStockout cost: economic consequences of running : economic consequences of running

out of stock (lost profit and/or goodwill)out of stock (lost profit and/or goodwill)

Economic Order Quantity Economic Order Quantity (EOQ)(EOQ)

Assumptions of the model:Assumptions of the model: Demand rate is constant, recurring and knownDemand rate is constant, recurring and known The lead time (from order placement and order The lead time (from order placement and order

delivery) is constant and knowndelivery) is constant and known No stockouts are allowedNo stockouts are allowed Goods are ordered and produced in lots, and the Goods are ordered and produced in lots, and the

lot is placed into inventory all at one timelot is placed into inventory all at one time Unit item cost is constant, carrying cost is linear Unit item cost is constant, carrying cost is linear

function of average inventory levelfunction of average inventory level Ordering cost is independent of the number of Ordering cost is independent of the number of

items in a lotitems in a lot Marginal holding cost is constantMarginal holding cost is constant The item is a single product (no interaction with The item is a single product (no interaction with

other products)other products)

The „SAW-TOOTH”The „SAW-TOOTH”Inventory level

Time

Orderquantity(Q)

Order inteval

Average inventory level = Q/2

Total cost of inventory Total cost of inventory (tradeoff between ordering frequency and (tradeoff between ordering frequency and

inventory level)inventory level)

Minimumcost

Total cost

Ordering cost

(S ∙ ∙ D/Q)

Holding cost

(H ∙∙ Q/2)

EOQ

Calculating the total costCalculating the total cost of of inventoryinventory

Let… Let… SS be the ordering cost (setup cost) per oder be the ordering cost (setup cost) per oder DD be demanded items per planning period be demanded items per planning period HH be the stock holding cost per unit be the stock holding cost per unit

HH==ii∙∙CC, where, where C C is the unit cost of an item,is the unit cost of an item, andand i i is the is the carrying ratecarrying rate

QQ be the ordered quantity per order (= lot) be the ordered quantity per order (= lot) TCTC = = SS ∙ ( ∙ (DD//QQ) + ) + HH ∙ ( ∙ (QQ/2)/2) ((DD//QQ) is the number of orders per period) is the number of orders per period ((QQ/2) is the average inventory level in this /2) is the average inventory level in this

modelmodel

The minimum The minimum costcost (EOQ) (EOQ)

TCTC = = SS ∙ ( ∙ (DD//QQ) + ) + HH ∙ ( ∙ (QQ/2)/2) ббTCTC//ббQQ = 0 = 0 0 = - 0 = - SS ∙ ( ∙ (DD//Q Q 22) + ) + HH/2/2 HH/2 = /2 = SS ∙ ( ∙ (DD//Q Q 22)) Q Q 2 2 = (2 ∙ = (2 ∙ SS ∙ ∙ DD)/H)/H EOQ = √ (2 ∙ EOQ = √ (2 ∙ SS ∙ ∙ DD)/H)/H

ExampleExampleDD = 1000 units per year = 1000 units per year

SS = 100 euro per order = 100 euro per order

HH = 20 euro per unit = 20 euro per unit

Find the economic order quantity! (we assume a saw-tooth model)

EOQEOQ = = √ (2 ∙ 1,000√ (2 ∙ 1,000unitsunits ∙ 100 ∙ 100euroeuro)/20)/20euro/uniteuro/unit

EOQEOQ = = √ 10,000√ 10,000unitsunits2 2 = = 100100unitsunits

The EOQ zoneThe EOQ zone

Reordering Reordering (or (or replenishment) replenishment) pointpoint

When to start the ordering process?When to start the ordering process? It depends on the…It depends on the…

Stock position: stock on-hand (+ stock on-order)Stock position: stock on-hand (+ stock on-order) in a simple saw-tooth model it is in a simple saw-tooth model it is Q, Q, in some cases, there can be an initial stockin some cases, there can be an initial stock ((QQ00)), , that is that is

different fromdifferent from Q. Q. In this case the first order depends on In this case the first order depends on QQ00

lead time (lead time (LTLT): the time interval from setting up ): the time interval from setting up order to the start of using up the ordered stockorder to the start of using up the ordered stock

Average demand per day (Average demand per day (dd)) ROPROP = = d d ((LTLT ) + safety stock ) + safety stock

ExamplesExamples QQ00 = 600 tons = 600 tons QQ = 200 tons = 200 tons dd = 10 tons per day = 10 tons per day LTLT = 8 days = 8 days What is the ROP and whenWhat is the ROP and when

we reach that level?we reach that level? What is the time of the next reorder?What is the time of the next reorder?

ROP = ROP = 80 tons80 tons

(600 – 80)/10(600 – 80)/10 = 52. day= 52. day

QQ00 = = QQ = 400 tons = 400 tons dd = 16 tons per day = 16 tons per day LTLT = 20 days = 20 days ROP = ?ROP = ? First rearder time?First rearder time?

ROP ROP = 16 ∙ 20 = 320 tons= 16 ∙ 20 = 320 tons

First reorder: (400First reorder: (400 – 320)/16 = – 320)/16 = 5. day5. day

52 + 8 + (200 – 80)/10 = 72. 52 + 8 + (200 – 80)/10 = 72. day day

ExamplesExamplesD D = 2,000 tons= 2,000 tonsS S = 100 euros per order= 100 euros per orderH H = 25 euros per order= 25 euros per orderInitial stockInitial stock = 1,000 tons = 1,000 tonsLTLT = 12 days = 12 daysNN = 250 days = 250 days

EOQEOQ = √ (2 ∙ 2,000 = √ (2 ∙ 2,000tsts ∙ 100 ∙ 100euroeuro)/25)/25euro/tseuro/ts = 126.49= 126.49tsts

dd = 2,000 = 2,000tsts/250/250ds ds = 8 = 8 ts/dsts/ds; ; ROPROP = 12 ∙ 8 = 96 tons = 12 ∙ 8 = 96 tons

ReroderReroder11 = (1,000 – 96)/8 = 113 = (1,000 – 96)/8 = 113

ReorderReorder2 2 = 113 + 12 + (126.49 – 96)/8 = 128.81 = = 113 + 12 + (126.49 – 96)/8 = 128.81 = 128128

Calculate the Calculate the following:following:

EOQEOQ

dd

ROPROP

first and second first and second reorder timereorder time

The SAW-TOOTHThe SAW-TOOTHwith safety stockwith safety stock

Inventory level

Time

Orderquantity

Continuous demand

Safety stock or buffer stock

b

Buffer stock dependsBuffer stock depends

Demand rate and lead timeDemand rate and lead time Variability of demand and lead timeVariability of demand and lead time Desired service levelDesired service level

Service levelService level

The probability that demand will not The probability that demand will not exceed supply during lead time.exceed supply during lead time.

Service level = 100 percent - Service level = 100 percent - stockout risk stockout risk

Buffer (safety) stockBuffer (safety) stock

b = zb = z ∙ σ ∙ σ

wherewhere

z = z = safety factor from the (normal) distributionsafety factor from the (normal) distribution

σ σ = = sandard deviation of demand over lead timesandard deviation of demand over lead time

LetLet z z be 1,65 (95%), and the standard be 1,65 (95%), and the standard deviation of demand is 200 units/lead deviation of demand is 200 units/lead

time.time.bb = 1,65 = 1,65 ∙∙ 200 200units units = 330= 330unitsunits

ExampleExample Lead time = 10 daysLead time = 10 days Average demand over lead time: 300 tonsAverage demand over lead time: 300 tons Standard deviation over lead time: 20 tonsStandard deviation over lead time: 20 tons Accepted risk level: 5%Accepted risk level: 5% Safety stock = ? Reorder quantity = ?Safety stock = ? Reorder quantity = ?

bb = = zz * * σσ = = 1,65 * 20 = 33 tons 1,65 * 20 = 33 tons ROP = 300 + 33 = 333 tonsROP = 300 + 33 = 333 tons

ExamplesExamples QQ00 = 600 tons = 600 tons QQ = 200 tons = 200 tons dd = 10 tons per day = 10 tons per day LTLT = 8 days = 8 days bb = 33 tons = 33 tons

ROP ROP = 8 * 10 + 33 = 113= 8 * 10 + 33 = 113

QQ00 = = QQ = 400 tons = 400 tons dd = 16 tons per day = 16 tons per day LTLT =20 days =20 days bb = 66 tons = 66 tons

ROP ROP = 386= 386

Economic production Economic production quantity (EPQ)quantity (EPQ)

Production of spare parts /materials done in Production of spare parts /materials done in batches (lots)batches (lots)

Only on itemOnly on item Annual demand is knownAnnual demand is known Usage rate (Usage rate (uu) is constant) is constant Usage occurs continually but production Usage occurs continually but production

occurs periodicallyoccurs periodically Production rate is constant (Production rate is constant (pp)) Lead time (Lead time (LTLT) does not vary) does not vary No quantity (No quantity (QQ) discounts) discounts Setup cost instead of ordering costSetup cost instead of ordering cost

EPQ with incremental EPQ with incremental inventory replenishmentinventory replenishment

Total cost in EPQTotal cost in EPQ

TCTCEPQEPQ = carrying cost + setup cost = = carrying cost + setup cost = = (= (IImaxmax/2)/2)HH + ( + (DD//QQ))SS

The economic run quantity:The economic run quantity:QQEPQEPQ = (2 = (2DSDS//HH))0,50,5 ∙ [ ∙ [pp/(/(p – up – u)])]0,50,5

Cycle time = Cycle time = QQEPQEPQ / / uu

Run time = Run time = QQEPQEPQ / / pp

IImax max = (= (QQEPQEPQ / / pp)) ((pp – – uu) )

IIaverage average = = IImax max / 2 / 2

EOQ exampleEOQ example

Quantity discountsQuantity discounts

Price reductions for large ordersPrice reductions for large orders Example:Example:

The buyers total cost curve to The buyers total cost curve to minimze:minimze:

AdvantagesAdvantages

For the buyerFor the buyer Fewer order set-ups (Fewer order set-ups (D*S/QD*S/Q)) Cheaper price (Cheaper price (P*DP*D))

For the seller:For the seller: Decreased holding costs (Decreased holding costs (I*H/2I*H/2)) Decreased administrative costs (FC)Decreased administrative costs (FC) Lower opportunity costLower opportunity cost

A TC görbe, ha A TC görbe, ha szerepeltetjük a beszerzési szerepeltetjük a beszerzési

költséget isköltséget is

AllowancesAllowances and TC and TC curvescurves

Constant carrying costsConstant carrying costs

Carrying costs are stated as Carrying costs are stated as a percentage of unit pricea percentage of unit price

Cost

Quantity

Finding EOQ with Finding EOQ with constant holding costconstant holding cost

DD = 816 pieces/year = 816 pieces/year SS = 12 dollars/order = 12 dollars/order HH = 4 dollars/piece/year = 4 dollars/piece/year PricesPrices: :

20 dollars 1-49 pieces, 20 dollars 1-49 pieces, 18 dollars 50-79 pieces, 18 dollars 50-79 pieces, 17 dollars 80-99 pieces, 17 dollars 80-99 pieces, 16 dollars over 100 pieces16 dollars over 100 pieces

Calculating EOQCalculating EOQ

SolutionSolution

QQEOQEOQ = (2*816*12/4) = (2*816*12/4)0,5 0,5 = 69.97 = 70 = 69.97 = 70 piecespieces

TCTC7070 = (70/2)*4 + (816/70)*12 + = (70/2)*4 + (816/70)*12 + 18*816 =18*816 == 14.968 dollars= 14.968 dollars

TCTC8080 = 14.154 = 14.154

TCTC100100 = 13.354 = 13.354

With non-constant With non-constant holding costsholding costs

DD = 4.000 pieces, = 4.000 pieces, SS = 30 dollars, = 30 dollars, HH = 0.4*= 0.4*PP

PricesPrices: : 1-499 pieces 0.9 dollar; 1-499 pieces 0.9 dollar; 500-999 0.85 dollar; 500-999 0.85 dollar; over 1.000 pieces 0.8 dollar.over 1.000 pieces 0.8 dollar.

SolutionSolution

1.1. QQEOQ(0,8)EOQ(0,8)=866 pieces =866 pieces → not feasible→ not feasible

2.2. QQEOQ(0,85)EOQ(0,85)=840 pieces =840 pieces → feasible→ feasible

3.3. TCTC840840=30*(4000/840)+0,4*0,85*(840/2)+4000*0=30*(4000/840)+0,4*0,85*(840/2)+4000*0,85=,85==3685,66 dollars=3685,66 dollars→ is it profitable to order a → is it profitable to order a greater lot?greater lot?

4.4. TCTC10001000=30*(4000/1000)+0,4*0,8*(1000/2)+4000=30*(4000/1000)+0,4*0,8*(1000/2)+4000*0,8=*0,8==3480 dollars=3480 dollars→ yes, it is profitable.→ yes, it is profitable.

Alternative models 1Alternative models 1

Periodic review system:Periodic review system: Stock level is examined at regular Stock level is examined at regular

intervalsintervals Size of the order depends on the quantity Size of the order depends on the quantity

on stock. it should bring the inventory to on stock. it should bring the inventory to a predetermined levela predetermined level

time

Stock on hand

L

Q Q

LLT TT

Q

Alternative models 2Alternative models 2

Fixed-order-quantity system:Fixed-order-quantity system: A predetermined stock level (reorder A predetermined stock level (reorder

point) is given, at which the point) is given, at which the replenishement order will be placedreplenishement order will be placed

The order quantity is constantThe order quantity is constant

R

Stock on hand

L LL

QQ

Elements of demand Elements of demand patterns (forecasting)patterns (forecasting)

Actual demand:Actual demand: Trend lineTrend line Seasonal fluctuacionSeasonal fluctuacion Weekly fluctuationWeekly fluctuation (Daily fluctuation)(Daily fluctuation) Random fluctuationRandom fluctuation

Inventory decisions and Inventory decisions and Multiple Multiple Distribution Distribution Centres / warehousesCentres / warehouses

The ‘square root law’:The ‘square root law’: A rule of thumbA rule of thumb The total safety-stock holding in a distribution system is The total safety-stock holding in a distribution system is

proportional to the square root of the number of depot proportional to the square root of the number of depot locations locations

Example: If we reduce the number of DCs from 10 to 5, the Example: If we reduce the number of DCs from 10 to 5, the savings in safety stock is:savings in safety stock is: 1 – (√5 / √10) = 29% 1 – (√5 / √10) = 29%

Pareto’s law Pareto’s law or the ’80/20 rule’:or the ’80/20 rule’: A rule of thumbA rule of thumb Approximately 20% of storage items account for 80% of Approximately 20% of storage items account for 80% of

the inventory value measured in money.the inventory value measured in money. ABC analysis (or Pareto analysis):ABC analysis (or Pareto analysis):

‘‘A’ lines: fast movers (20%) – 80% of money usageA’ lines: fast movers (20%) – 80% of money usage ‘‘B’ lines: medium movers (30%) – 15% of money B’ lines: medium movers (30%) – 15% of money

usageusage ‘‘C’ lines: slow movers (C+D 50%) – 5% of money C’ lines: slow movers (C+D 50%) – 5% of money

usageusage ‘‘D’ lines: obsolete / dead stockD’ lines: obsolete / dead stock

ExampleExample

Initial number of warehouses: 6Initial number of warehouses: 6 Initial sum of safety stocks: 6,000Initial sum of safety stocks: 6,000 New number of warehouses: 2New number of warehouses: 2 What is the new sum of safety stocks? What is the new sum of safety stocks?

6000*(2/6)0.5=3464.10

ABC analysisABC analysis

ABC analysisABC analysis

ABC analysis exerciseABC analysis exerciseItem Item numbernumber

Annual demandAnnual demand PricePrice Annual dollar valueAnnual dollar value

11 2.5002.500 360360 ??

22 1.0001.000 7070 ??

33 2.4002.400 500500 ??

44 1.5001.500 100100 ??

55 700700 7070 ??

66 1.0001.000 10001000 ??

77 200200 210210 ??

88 1.0001.000 40004000 ??

99 8.0008.000 1010 ??

1010 500500 200200 ??

??

ABC analysis exerciseABC analysis exerciseItem Item numbernumber

Annual demandAnnual demand PricePrice Annual dollar valueAnnual dollar value

11 2.5002.500 360360 900.000900.000

22 1.0001.000 7070 70.00070.000

33 2.4002.400 500500 1.200.0001.200.000

44 1.5001.500 100100 150.000150.000

55 700700 7070 49.00049.000

66 1.0001.000 10001000 1.000.0001.000.000

77 200200 210210 42.00042.000

88 1.0001.000 40004000 4.000.0004.000.000

99 8.0008.000 1010 80.00080.000

1010 500500 200200 100.000100.000

∑ ∑ 7.591.0007.591.000

SolutionSolutionItem Item

numbernumberAnnual dollar Annual dollar ClassificationClassification Percentage of Percentage of

itemsitemsPercentage of Percentage of annual dollar annual dollar

valuevalue

88 4.000.0004.000.000

33 1.200.0001.200.000

66 1.000.0001.000.000

11 900.000900.000

44 150.000150.000

1010 100.000100.000

99 80.00080.000

22 70.00070.000

55 49.00049.000

77 42.00042.000

SolutionSolutionItem Item

numbernumberAnnual dollar Annual dollar ClassificationClassification Percentage of Percentage of

itemsitemsPercentage of Percentage of annual dollar annual dollar

valuevalue

88 4.000.0004.000.000 AA 1010 52,752,7

33 1.200.0001.200.000 BB

3030 40,840,866 1.000.0001.000.000 BB

11 900.000900.000 BB

44 150.000150.000 CC

6060 6,56,5

1010 100.000100.000 CC

99 80.00080.000 CC

22 70.00070.000 CC

55 49.00049.000 CC

77 42.00042.000 CC

?

?

Data CaptureData Capture Techniques and error rates Techniques and error rates (Rushton et al. 2006)(Rushton et al. 2006)

Written entry – 25,000 in 3,000,000Written entry – 25,000 in 3,000,000 Keyboard entry – 10,000 in 3,000,000Keyboard entry – 10,000 in 3,000,000 Optical character recognition (OCR) – 100 in 3,000,000Optical character recognition (OCR) – 100 in 3,000,000

labels that are both machine- and human-readablelabels that are both machine- and human-readable for example: license platesfor example: license plates

Bar code (code 39) – 1 in 3,000,000Bar code (code 39) – 1 in 3,000,000 fast, accurate and fairly robust fast, accurate and fairly robust reliable and cheap techniquereliable and cheap technique

Transponders (radio frequency tags) – 1 in 30,000,000Transponders (radio frequency tags) – 1 in 30,000,000 a tag (microchip + antenna) affixed to the goods or containera tag (microchip + antenna) affixed to the goods or container receiver antennareceiver antenna readerreader host station that relays the data to the serverhost station that relays the data to the server can be passive or activecan be passive or active

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