1

Supply Chain Management

Traditional Inventory Models for Independent Demand

Prof. Prem Vrat

Vice Chancellor

UP Technical University

Lucknow

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What is an Inventory System

Inventory is defined as the stock of any item or resource used in an organization.

An Inventory System is made up of a set of policies and controls designed to monitor the levels of inventory and designed to answer the following questions: What levels should be maintained? When stock should be replenished? and How large orders should be? i.e. what is the

optimal size of the order?

3

Cost of holding inventory

Capital cost (Interest) Storage cost Obsolescence cost Consist of about 20% of total cost in the

United States

4

Inventory issues

Demand Constant vs. variable deterministic vs. stochastic

Lead time Review time

Continuous vs. periodic Excess demand

Backorders, lost sales Inventory change

Perish, obsolescence

Inventory Decisions: When, What, and how many to order

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Two basic types of Inventory Systems

1) continuous (fixed-order quantity) an order is placed for the same constant amount

when inventory decreases to a specified level, ie. Re-order point

2) periodic (fixed-time) an order is placed for a variable amount after a

specified period of time used in smaller retail stores, drugstores, grocery

stores and offices

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basic inventory elements

1.1. Carrying cost, Carrying cost, Cc

• Include facility operating costs, record keeping, interest, etc.

2.2. Ordering cost, Ordering cost, Co

• Include purchase orders, shipping, handling, inspection, etc.

3.3. Shortage (stock out) cost, Shortage (stock out) cost, Cs

• Sometimes penalties involved; if customer is internal, work delays could result

Carrying Costs

Category

Cost (and Range) as a Percent of Inventory

Value

Housing costs (including rent or depreciation, operating costs, taxes, insurance)

6% (3 - 10%)

Material handling costs (equipment lease or depreciation, power, operating cost)

3% (1 - 3.5%)

Labor cost 3% (3 - 5%)

Investment costs (borrowing costs, taxes, and insurance on inventory)

11% (6 - 24%)

Pilferage, space, and obsolescence 3% (2 - 5%)

Overall carrying cost 26%

8

Inventory

- to study methods to deal with

“how much stock of items should be kept on hands that would meet customer demand”

Objectives are to determine:

a) how much to order, and

b) when to order

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Inventory models

Here, we onlyonly study the following three different models:

1. Basic model

2. Model with “discount rate”

3. Model with “re-order points”

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1. Basic model

The basic model is known as:

“Economic Order Quantity” (EOQ) Models

Objective is to determine the optimal order size that will minimize total inventory costs

How the objective is being achieved?

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Quantityon hand

Q

Receive order

Placeorder

Receive order

Placeorder

Receive order

Lead time

Reorderpoint

Usage rate

Profile of Inventory Level Over Time

12

Profile of … Frequent Orders

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Basic EOQ models

Three models to be discussed:

1. Basic EOQ model

2. EOQ model without instantaneous

receipt

3. EOQ model with shortages.

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Basic Fixed-Order Quantity Model

This model attempts to estimate the order size (Q) and determine the point (R) at which an order should be placed.

Model assumptions:

1. Annual demand (D) for the product is known, constant and uniform throughout the period,

2. Lead time (L) is known and constant,

3. Product unit price (C) is known and constant,

4. Per unit holding or carrying cost (Cc) is known and constant,

5. Ordering or setup cost (Co) is known and constant,

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Basic Fixed-Order Quantity Model

Model assumptions:

6 No backorders are allowed,

7. There is no interaction with other products, the inventory control system operates independent of its environment.

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1 Economic Order Quantity (EOQ) Model

QOrder Size

L Time

Q/2

d

L

R

Daily Usage Rate

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The Basic EOQ Model

• The optimal order size, Q, is to minimize the sum of carrying costs and ordering costs.

• Assumptions and Restrictions: - Demand is known with certainty and is relatively constant over time. - No shortages are allowed. - Lead time for the receipt of orders is constant. (will consider later) - The order quantity is received all at once and instantaniously.

How to determinethe optimal valueQ*?

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Determine of Q

We try to Find the total cost that need to spend for keeping

inventory on hands = total ordering + stock on hands Determine its optimal solution by finding its first

derivative with respect to Q

How to get these values?1. Find out the total carrying cost2. Find out the total ordering cost3. Total cost = 1 + 24. d (Total cost) /d Q = 0, and find Q*

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The Basic EOQ ModelWe assumed that, we will only keep half the inventory over a year then

The total carry cost/yr = Cc x (Q/2). Total order cost = Co x (D/Q)

Then , Total cost = 2QC

QDCTC co Finding optimal Q*

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The Basic EOQ Model

• EOQ occurs where total cost curve is at minimum value and carrying cost equals ordering cost:

•Where is Q* located in our model?

c

o

co

CDCQ

QCQDCTC

2

2

*

min

(How to obtain this?)Then, *c

o

co

CDCQ

QCQDCTC

2

2

*

min

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EOQ Derivation

Min. The Total Cost Function by finding first derivative and equating it to zero:

TC = DCo/Q + QCc/2

dTC

Solving for Q: EOQ =

We could achieve the same result by equating Holding (Carrying Cost) to Ordering Cost and solve for Q .

Reorder point: R = d.L + SS (safety stock)

dQ = (- DCo/Q ) + Cc/2 = 0

2

2DCoCc

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The Basic EOQ Model

• Total annual inventory cost is sum of ordering and carrying cost:

2QC

QDCTC co

Figure The EOQ cost model

To order inventory

To keep inventory

Try to get this value

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The Basic EOQ ModelExample

Consider the following:

days store 62.2 5

311*/

days 311 timecycleOrder

5000,2000,10 :yearper orders ofNumber

500,1$2

)000,2()75.0(000,2000,10)150(

2 :costinventory annual Total

yd 000,2)75.0(

)000,10)(150(22* :sizeorder Optimal

10,000yd D $150, C $0.75, C :parameters Model

*

*min

oc

QD

QD

QCQDCTC

CDCQ

optco

c

o

No. of working days/yr

*

Note: You should pay attention thatall measurement units must be the same

Consider the same example, with yearly

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The Basic EOQ ModelEOQ Analysis with monthly time frame

$1,500 ($125)(12) cost inventory annual Total

monthper 125$2

)000,2()0625.0(000,2

)3.833()150(2*

* :costinventory monthly Total

yd 000,2)0625.0(

)3.833)(150(22* :sizeorder Optimal

monthper yd 833.3 D order,per $150 C month,per ydper $0.0625 C :parameters Model

min

oc

QCQDCTC

CDCQ

co

c

o

(unit be based on yearly)

12 months a year

Robust Model

The EOQ model is robustThe EOQ model is robust

It works even if all parameters It works even if all parameters and assumptions are not metand assumptions are not met

The total cost curve is relatively The total cost curve is relatively flat in the area of the EOQflat in the area of the EOQ

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2 Fixed-Order Quantity Model With Usage

In this model production and usage of the item being manufactured occur simultaneously. The graph below illustrates the model.

TC = DC + DS/Q +

Production Rate, p

Usage Rate, d

R

Q

L

Build up

Usage Rate

(p-d)

No production

usage only

Max.

(p - d).Q.H 2p

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The EOQ Model with Noninstantaneous Receipt

Figure The EOQ model with noninstantaneous order receiptAlways greater than 0why?

The order quantity is received gradually over time and inventory is drawn on

at the same time it is being replenished.

Example: Let p = production, d = demand,

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The EOQ Model with Noninstantaneous ReceiptModel Formulation

)/1(2 :sizeorder Optimal

12

:cost inventory annual Total

demanded isinventory at which ratedaily drate) productionor ( over time received isorder heat which t ratedaily p

*

pdCDCQ

pdQC

QDCTC

c

o

co

Assuming placing an order/yr

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The EOQ Model with finite-replenishment rate

yd 772,1150

2.3218.256,21* levelinventory Maximum

runs 43.48.256,2

000,10*

runs)n (productioyear per orders ofNumber

days 05.15150

8.256,2* length run Production

329,1$150

2.3212

)8.256,2()075(.)8.256,2()000,10()150(1

2*

*min:costinventory annual minimum Total

yd 8.256,2

1502.32175.0

)000,10)(150(2

1

2* :sizeorder Optimal

dayper yd 150 p day,per yd 32.2 10,000/311 year,per yd 10,000 D unit,per $0.75 Cc $150, Co

pdQ

QD

pQ

pdQCc

QDCoTC

pdCc

CoDQ

Let,

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The EOQ Model with Shortages

• In the EOQ model wth shortages, the assumption that shortages cannot exist is relaxed.

• Assumed that unmet demand can be backordered with all demand eventually satisfied.

Shortage = S/Q

On hand = (Q-S)/Q t1 + t2 = S/D + (Q-S)/D = Q/D

Shortage

What we needed

Max level of inventory

Here, we allow Q being shortageshortage, so that we could borrow or replenish the stockslater

Total cost is

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3 The EOQ Model with Shortages

• In the EOQ model wth shortages, the assumption that shortages cannot exist is relaxed.

• Assumed that unmet demand can be backordered with all demand eventually satisfied.

Shortage = S/Q

On hand = (Q-S)/Q t1 + t2 = S/D + (Q-S)/D = Q/D

Shortage

What we needed

½*base* height = ½ * (Q-S) * (Q-S)/Q = ½ * (Q-S)2 /Q

Area = ½ * (S/Q) * S = ½ * S2 /Q

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The EOQ Model with Shortages

sc

c

s

cs

c

o

ocs

CCCQS

C

CC

C

DCQ

Q

DC

Q

SQC

Q

SCTC

** :level Shortage

2*:quantityorder Optimal

2

)(

2:costinventory Total

cost ordering total costs carrying total costs shortage Total cost Total

22

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The EOQ Model with ShortagesExample

$1,279.20 639.60 465.16 $174.44

2.345,2)000,10)(150(

)2.345,2(2)6.705,1)(75.0(

)2.345,2(2)6.639)(2(

**2*)*(

*2:costinventory Total

yd 6.63975.02

75.02.345,2* :level Shortage

yd 2.345,22

75.0275.0

)000,10)(150(22:quantityorder Optimal

yd 10,000 D yd,per $2 C yd,per $0.75 C $150, C

2222

*

*

sco

QDC

QSQC

QSCTC

CCCQS

CCC

CDCQ

ocs

sc

c

s

cs

c

o

Let,

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The EOQ Model with Shortages

Additional Parameters in Example

sday 19.9or year 0.064 10,000639.6

D t shortage a is re which theduring Time

days 53.2or 0.17110,000

639.6-2,345.2 t handon isinventory which during Time

ordersbetween days 0.7326.4

311

orders ofnumber

yearper days t ordersbetween Time

yd 6.705,16.6392.345,2 levelinventory Maximum

yearper orders 26.42.345,2

000,10 orders ofNumber

2

1

SD

SQ

SQ

QD

= Q/D

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4. Model with “discount rate”

Price discounts are often offered if a predetermined number of units is ordered or when ordering materials in high volume.

How do we decide if we should order more to take advantage of the discount being offered?

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All-Unit Quantity Discounts

Pricing schedule has specified quantity break points q0, q1, …, qr, where q0 = 0

If an order is placed that is at least as large as qi but smaller than qi+1, then each unit has an average unit cost of Ci

The unit cost generally decreases as the quantity increases, i.e., C0>C1>…>Cr

The objective for the company (a retailer for example) is to decide on a lot size that will minimize the sum of material, order, and holding costs

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All-Unit Quantity Discount Procedure

Step 1: Calculate the EOQ for the lowest price. If it is feasible (i.e., this order quantity is in the range for that price), then stop. This is the optimal lot size. Calculate TC for this lot size.

Step 2: If the EOQ is not feasible, calculate the TC for this price and the smallest quantity for that price.

Step 3: Calculate the EOQ for the next lowest price. If it is feasible, stop and calculate the TC for that quantity and price.

Step 4: Compare the TC for Steps 2 and 3. Choose the quantity corresponding to the lowest TC.

Step 5: If the EOQ in Step 3 is not feasible, repeat Steps 2, 3, and 4 until a feasible EOQ is found.

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All-Unit Quantity Discounts: Example

Cost/Unit

$3$2.96

$2.92

Order Quantity

5,000 10,000

Order Quantity

5,000 10,000

Total Material Cost

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All-Unit Quantity Discount: Example

Order quantity Unit Price0-5000 $3.005001-10000 $2.96Over 10000 $2.92

q0 = 0, q1 = 5000, q2 = 10000C0 = $3.00, C1 = $2.96, C2 = $2.92D = 120000 units/year, Co = $100/lot, Cc = 0.2

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All-Unit Quantity Discount: Example

Step 1: Calculate Q2* = Sqrt[(2DCo)/CcC2] = Sqrt[(2)(120000)(100)/(0.2)(2.92)] = 6410Not feasible (6410 < 10001)Calculate TC2 using C2 = $2.92 and q2 = 10001TC2 = (120000/10001)(100)+(10001/2)(0.2)(2.92)+ (120000)(2.92)

= $354,520Step 2: Calculate Q1* = Sqrt[(2DCo)/CcC1]=Sqrt[(2)(120000)(100)/(0.2)(2.96)] = 6367Feasible (5000<6367<10000) StopTC1 = (120000/6367)(100)+(6367/2)(0.2)(2.96)+ (120000)(2.96) =

$358,969TC2 < TC1 The optimal order quantity Q* is q2 = 10001

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All-Unit Quantity Discounts

What is the effect of such a discount schedule? Retailers are encouraged to increase the size of

their orders Average inventory (cycle inventory) in the supply

chain is increased Average flow time is increased Is an all-unit quantity discount an advantage in the

supply chain?

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5. Model with “re-order points”

• The reorder point is the inventory level at which a new order is placed.

• Order must be made while there is enough stock in place to cover demand during lead time.

• Formulation: R = dL, where d = demand rate per time period, L = lead time

Then R = dL = (10,000/311)(10) = 321.54

Working days/yr

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Reorder Point• Inventory level might be depleted at slower or faster rate during lead time.

• When demand is uncertain, safety stock is added as a hedge against stockout.

Two possible scenarios

Safety stock!

No Safetystocks!

We should then ensureSafety stock is secured!

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Determining Safety Stocks Using Service Levels

• We apply the Z test to secure its safety level,

)( LZLdR d

Reorder point

Safety stock

Average sample demand

How these values are represented in the diagram of normal distribution?

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Reorder Point with Variable Demand

stocksafety

yprobabilit level service toingcorrespond deviations standard ofnumber demanddaily ofdeviation standard the

timeleaddemanddaily average

pointreorder where

LZ

Z

Ld

R

LZLdR

d

d

d

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Reorder Point with Variable DemandExample

Example: determine reorder point and safety stock for service level of 95%.

26.1. : formulapoint reorder in termsecond isstock Safety

yd 1.3261.26300)10)(5)(65.1()10(30

1.65 Zlevel, service 95%For

dayper yd 5 days, 10 L day,per yd 30 d

LZLdR

d

d