OPSM 301 Operations Management

Class 15:

Inventory Management

EOQ Model

Koç University

Zeynep Aksinzaksin@ku.edu.tr

InventoryInventory

“The stock of any item or resource used in an organization”

“All the money that the system has invested in purchasing things it intends to sell”

The Material Flow Cycle

Types of Inventories

Inputs - Raw Materials Processes - Work-in-Progress Outputs - Finished Goods

Why do we need Inventory?

Variability (uncertainty)– Demand– Capacity availability– Materials and lead times– Processing times

Time– Delivery lead time, production lead time

Economies of Scale– Purchasing, production

Functions Provided by Inventories

Purpose /Reason Type Cost

Transportation Pipeline Transportation Costs

Economies in Setups Cycle Stocks Setup/Order Costs

Seasonality in Demand Seasonal Stock Smoothing Costs

Uncertainty in Demand Safety Stock Shortage/Stock-out

Costs

Economies in Purchase Cycle Stocks Price Discounts

Inflation and/or Price Fluctuations Speculative Stock Costs due to Price

Inventory Costs

Purchase Cost Ordering Cost

– Receiving and inspection– Transportation

Holding (Carrying) Cost– Cost of money– Insurance– Taxes– Shrinkage, spoilage, obsolescence

Stock-out (Shortage) Cost– Lost sales, customers etc.– Emergency shipment costs

Economies of Scale:Inventory Management for a Retailer

The South Face retail shop in the John Hancock Tower has observed a stable monthly demand for its line of Gore-Tex jackets on the order of 100 jackets per month. The retail shop incurs a fixed cost of $2,000 every time it places an order to the Berkeley warehouse for stock replenishment. The marginal cost of a jacket is $200, and South Face’s cost of capital is approximately 25%.

What order size would you recommend for The South Face?

retailerwarehouse

Parameters EOQ Model

D demand rate (units per year)C unit production cost, not

counting setup or inventory costs (dollars per unit)

S fixed or setup cost to place an order (dollars)

H holding cost (dollars per year); if the holding cost is consists entirely

of interest on money tied up in inventory, then H = iC where i is an annual interest rate.

Q the unknown size of the order or lot size

Inventory Usage Over Time

Time

Inve

ntor

y Le

vel

AverageInventory

(Q*/2)

0Minimum inventory

Order quantity = Q (maximum inventory level)

Usage Rate

Cost Minimization GoalCost Minimization Goal

H2Q

S + QD

TC = DC +

Ordering Cost

Holding Cost

QOPT Order Quantity (Q)

COST

Annual Cost ofItems

Total Cost (TC)

H2Q

S + QD

TC = DC +

Total Annual CostTotal Annual Cost

Total Annual Cost = Annual

PurchasingCost

AnnualOrdering

Cost

AnnualHolding

Cost+ +

Using calculus, we can take the derivative of the total cost function and set the derivative (slope) equal to zero

We can also use economic intuition

Find most economical order quantity: Spreadsheet for The South Face

Number of units Number ofper order/batch Batches per Annual Annual Annual

Q Year: R/Q Setup Cost Holding Cost Total Cost

50 24 48000 1250 49250100 12 24000 2500 26500150 8 16000 3750 19750200 6 12000 5000 17000250 5 9600 6250 15850260 5 9231 6500 15731270 4 8889 6750 15639280 4 8571 7000 15571290 4 8276 7250 15526300 4 8000 7500 15500310 4 7742 7750 15492320 4 7500 8000 15500330 4 7273 8250 15523340 4 7059 8500 15559350 3 6857 8750 15607400 3 6000 10000 16000500 2 4800 12500 17300600 2 4000 15000 19000700 2 3429 17500 20929

Deriving the EOQDeriving the EOQ

EOQ Model: if there is a lead time LEOQ Model: if there is a lead time LEOQ Model: if there is a lead time LEOQ Model: if there is a lead time L

ROP = Reorder point L = Lead time (constant)Q = Economic order quantity

L L

ROP

Time

# U

nit

s on

han

dQopt

EOQ ExampleEOQ Example

• Annual Demand = 1,000 units• Days per year considered in average daily demand = 250• Cost to place an order = $10• Holding cost per unit per year = $0.50• Lead time = 7 days• Cost per unit = $15

Determine the economic order quantity and the reorder point

An EOQ Example

Determine optimal number of needles to orderDetermine optimal number of needles to orderD D = 1,000= 1,000 units unitsS S = $10= $10 per order per orderH H = $.50= $.50 per unit per year per unit per year

Q* =Q* =22DSDS

HH

Q* =Q* =2(1,000)(10)2(1,000)(10)

0.500.50= 40,000 = 200= 40,000 = 200 units units

An EOQ Example

Determine optimal number of needles to orderDetermine optimal number of needles to orderD D = 1,000= 1,000 units units Q* Q* = 200= 200 units unitsS S = $10= $10 per order per orderH H = $.50= $.50 per unit per year per unit per year

= N = == N = =Expected Expected number of number of

ordersorders

DemandDemandOrder quantityOrder quantity

DDQ*Q*

N N = = 5= = 5 orders per year orders per year 1,0001,000200200

An EOQ Example

Determine optimal number of needles to orderDetermine optimal number of needles to orderD D = 1,000= 1,000 units units Q*Q* = 200= 200 units unitsS S = $10= $10 per order per order NN = 5= 5 orders per year orders per yearH H = $.50= $.50 per unit per year per unit per year

= T == T =Expected Expected

time between time between ordersorders

Number of working Number of working days per yeardays per year

NN

T T = = 50 = = 50 days between ordersdays between orders250250

55

An EOQ Example

Determine optimal number of needles to orderDetermine optimal number of needles to orderD D = 1,000= 1,000 units units Q*Q* = 200= 200 units unitsS S = $10= $10 per order per order NN = 5= 5 orders per year orders per yearH H = $.50= $.50 per unit per year per unit per year TT = 50= 50 days days

Total annual cost = Setup cost + Holding costTotal annual cost = Setup cost + Holding cost

TC = S + HTC = S + HDDQQ

QQ22

TC TC = ($10) + ($.50)= ($10) + ($.50)1,0001,000200200

20020022

TC TC = (5)($10) + (100)($.50) = $50 + $50 = $100= (5)($10) + (100)($.50) = $50 + $50 = $100

Reorder 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)Figure 12.5Figure 12.5 Lead time = LLead time = L

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

Reorder Point Example

Demand Demand = 8,000= 8,000 iPods per year iPods per year250250 working day year working day yearLead time for orders is Lead time for orders is 33 working days working days

ROP =ROP = d x L d x L

d =d = DD

Number of working days in a yearNumber of working days in a year

= 8,000/250 = 32= 8,000/250 = 32 units units

= 32= 32 units per day x units per day x 33 days days = 96= 96 units units

Economic Order Quantity (EOQ) Model

Economic Order Quantity (EOQ) Model– Robust, widely used

– Insensitive to errors in estimating parameters (40-20-2 Rule):

• 40% error in one of the parameters

• 20% error in Q

•< 2% of total cost penalty

An EOQ Example

Management underestimated demand by 50%Management underestimated demand by 50%D D = 1,000= 1,000 units units Q*Q* = 200= 200 units unitsS S = $10= $10 per order per order NN = 5= 5 orders per year orders per yearH H = $.50= $.50 per unit per year per unit per year TT = 50= 50 days days

TC = S + HTC = S + HDDQQ

QQ22

TC TC = ($10) + ($.50) = $75 + $50 = $125= ($10) + ($.50) = $75 + $50 = $1251,5001,500200200

20020022

1,500 1,500 unitsunits

Total annual cost increases by only 25%Total annual cost increases by only 25%

An EOQ Example

Actual EOQ for new demand is Actual EOQ for new demand is 244.9244.9 units unitsD D = 1,000= 1,000 units units Q*Q* = 244.9= 244.9 units unitsS S = $10= $10 per order per orderH H = $.50= $.50 per unit per year per unit per year

TC = S + HTC = S + HDDQQ

QQ22

TC TC = ($10) + ($.50)= ($10) + ($.50)1,5001,500244.9244.9

244.9244.922

1,500 1,500 unitsunits

TC TC = $61.24 + $61.24 = $122.48= $61.24 + $61.24 = $122.48