Graduate Program in Business Information Systems

Inventory Decisions with Certain Factors

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A Retailer’s Plea

If I order too little, I make no profit. If I order too much, I may go broke. Every product is different. Help me!

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Why do we control inventory? Inventories represent a vast segment of total

economic activity. Even minor improvements can create large

savings.

How do we control inventory? Application of optimization techniques Information processing and retrieval

techniques

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Decisions of an inventory policy If there is no production, i.e., pure inventory

system How much to order? Order quantity When to order? Reorder quantity

Ex:Order Q=100 units when the inventory level drops to r=15 units.

If there is also production When to start/stop production?

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An inventory system

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Elements of Inventory Decisions Costs:

Ordering and Procurement costs Inventory holding or carrying costs Inventory shortage costs

Demand structure How does it vary? Certain, uncertain?

Supply structure Any capacity limitations, defectives, number of suppliers?

Lead times: Certain, uncertain?

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Ordering and Procurement Costs Represent all expenses incurred in ordering or

manufacturing items related to Acquisition Transportation Collecting, sorting, placing the items in the storage Managerial and clerical costs associated with order

placement. Ordering costs are fixed, independent of the order size. Procurement costs depend on the order size.

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Holding or Carrying Costs Expenses incurred during the storage of items.

Physical Costs: Warehouse operation costs, insurence, property taxes.

Pilferage, spoilage, obsolescence Opportunity cost of investing in inventory rather

than investing somewhere else, ex. in a bank. Inventory costs are variable costs that depend

on the order size.

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Shortage Costs Occur whenever the demand is not satisfied.

Order is either “backordered” or “lost”. Backordering Costs:

Fixed cost of extra managerial work. Loss of customer goodwill: Variable cost that

depends on duration of backorder. Lost Sales Costs:

Marginal profit that the item would have earned. Loss of customer goodwill.

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Demand Structure Continuous versus discrete demand

Ex: Natural gas consumption in housesDetergent consumption in houses

Deterministic (certain) versus stochastic (uncertain) demandEx: Order quantities for the next months are 20,30,10,50.

Order quantities in a month are normally distributed with mean 25 and variance 4.

Constant versus dynamic demandEx: Demand quantities for the next months are 20, 21, 20, 19

Demand quantities for the next months are 20, 50, 10, 2

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Supply Structure Any defectives?

If the received lot includes defective items this brings uncertainty

Any capacity limitations?

Do we fully receive what we order? Number of suppliers, fixed or variable?

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Lead time Time elapsed between the order delivery and

order receipt. Can be constant or stochastic.

Ex: Lead time is 10 days.

Lead time is between 8-12 days.

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The Economic Order QuantityEOQ-Model

Decision variable: Q = Order Quantity Parameters:

k = Fixed cost per order ($/order)

A = Annual number of items demanded (unit/year)

c = Unit cost of procuring an item ($/unit)

h = Annual cost of holding a dollar in inventory ($/$/year) Objective is to “minimize total annual cost”.

Total=

Ordering+

Holding+

ProcurementAnnual cost Cost Cost Cost

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EOQ Inventory Policy

Average Inv. Level

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Assumptions of Classical EOQ Model Demand rate is constant or stable. There is infinite supply availability. Lead time is constant or zero.

No quantity discounts are made. All demand is met on time, no backordering, no

stockout.

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Costs of EOQ Model Total ordering cost is the number of orders times

the cost per order:

Total holding cost is the cost per item held 1 year times the average inventory:

The annual procurement cost is the product of annual demand and unit cost:

Procurement cost = Ac

kQA

cost ordering Annual

2cost holding Annual

Qhc

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Annual Cost of EOQ-Model

Here Ac is not a relevant cost and thus dropped.

Minimize Total Annual Inventory Cost:

AcQ

hckQA

2 cost annual Total

2 )(

Qhck

QA

QTC

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Optimal Solution of EOQ Optimal solution is the economic order quantity

Optimal Total Cost hc

AkQ

2 *

AkhcTC 2*

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Example:The House of Wines and Liquors

Allex Mullen decides that the first task in utilizing inventory models is to determine the value of model parameters:

Annual demand 5200 cases of beer $10 telephone charge for ordering Purchase cost is $1.5/case beer +shipping cost

$0.5/case 10%bank interest, 5%state franchise tax, 5% theft

insurance rateHow many should he order, how often, and at what annual relevant inventory cost?

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Solution:The economic order quantity is

The inventory cycle duration is T = Q/A = 510/5200 = 0.098 year or 36 days

The total annual relevant inventory cost is:

510or9.509220.

10520022 *

hc

AkQ

yearTC /96.203$00.10296.101$2

510)2(20.10

510

5200 (510)

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Robustness of EOQ Model EOQ is a robust model with respect to the

estimation errors in A, k, c or h. Let Aactual=4 Aestimated

Then EOQactual=2 Aestimated

Since

estimatedestimatedactual

actual EOQhc

kA

hc

kAEOQ 2

22

2

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Ex: The House of Wines and Liquors Alex Mullen applies EOQ to another product,

a particular variety of Chilean wine that sells 1000 cases annually. The cost is $20 per case. A telephone call to Chile to place an order costs $100. The holding costs are the same as for Tres Equis Beer.

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Ex:

224or 6.2232020.

100100022 *

hc

AkQ

T = Q/A = 24/1000 = .224 year or 82 days

ar$894.43/ye2

224)20(20.100

224

1000 (224)

TC

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Optimal Inventory Policywith BackorderingOrders placed during shortages are backordered.

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Optimal Inventory Policywith BackorderingS: Quantity on hand when a shipment arrives.P: Cost of being one item short for a year

Optimal order quantity and order level:

Q

SQpQ

hcSk

QA

S,QTC22

)(22

hcpp

hcAk

*Sp

hcphcAk

*Q

2

2

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Example:The House of Wines and Liquors-BackordersThe marketing department tells Alex that beer is a convenience product that can not be backordered, so sale is lost! However some wine customers are connoisseurs who are willing to order out-of-stock items. Nevertheless, the store owner will incur some penalty cost if there is a shortage of wine.

Suppose that retailer suffers lost profit on future business equal to $0.01/unit each day that a wine is on backorder. What should be the optimal ordering policy if backordering is allowed?

Solution: The order quantity is computed:

p = $.01×365 = $3.65/unit/year.

324

65.3

2020.65.3

2020.

100100022 *

p

hcp

hc

AkQ

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The order level S is

The relevant cost is

smaller than before, why?

154

2020.65.3

65.3

2020.

100100022 *

hcp

p

hc

AkS

82.6173242

17065.3

3242

1542020.100

324

1000 )154,324(

22

TC

Example: Solution

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Is backordering better? Fewer orders are placed when there is backordering. Average inventory level is smaller.

Backorders/cycle= Q* – S*=324 – 154 = 170 units/cycle.Proportion of demand not satisfied on time

=(Q*-S*)/Q*=170/324= 52.5%

The results suggest that:Retailers will run short in each cycle. But can they get away with it?

So backordering must make sense!

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Imputed Shortage PenaltyAn alternative approach for establishing an inventory policy is based on achieving a desired service level.

Service Level, L is the proportion of demand met on time

Imputed shortage penalty

p =hcL

1 L

S so ,1 ***

**

LQLQ

SQ

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As p increases EOQ is more robust

$36imputed shortage penalty

236

212

L=90%

EOQ with no backordering

P

324

154

224

L=47.5%

Q*

S*

$3.65

A=1000 units/yr k=$100/order c=$20/unit h= $0.20/$/year

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Economic Production-Quantity Model

The inventory model may be extended to finding the optimal production quantity.

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B: Annual production rate K: Production setup cost. c: Variable production cost per unit. Total Annual Cost:

Economic Production Quantity:

B

ABQhck

Q

AQTC

2 )(

BAB

hcAk

*Q 2

Economic Production-Quantity Model

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Example:Water Wheelies have annual demand of A =100,000 units and are made at the rate of B = 500,000 units. Production costs are k = $2,000/setup and c = $5/unit variable. It costs h = $.40/year to tie up a dollar.

Economic production quantity is

Total relevant cost is

TC(8,944) 56.516,29$000,500

000,100000,500

2

944,8540.000,2

944,8

000,100

units944,8500

100500

540.

210022 *

B

AB

hc

AkQ

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More Elaborate Models Incorporate a second one-time shortage penalty. Add additional products. Incorporate uncertainty regarding:

Demand Lead-time for delivery of order

Incorporate lost sales Extend to single period products

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Economic Order Quantity Model(Figure 15-3)

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10111213141516

A B C D E F G H I

PROBLEM: House of Fine Wines and Liquors - Tres Equis Beer

Parameter Values:Fixed Cost per Order: k = 10.00$ Annual Number of Items Demanded: A = 5,200 Unit Cost of Procuring an Item: c = 2.00$ Annual Holding Cost per Dollar Value: h = 0.20$

Decision Variables:Order Quantity: Q = 100

Results:Total Annual Relevant Cost: TC = 540.00$ Time Between Orders (years): T = 0.0192

INVENTORY ANALYSIS - ECONOMIC ORDER QUANTITY MODEL

1516

F=(F7/F12)*F6+F9*F8*(F12/2)=F12/F7

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Sensitivity Analysis(Figure 15-6)

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A B C D E F G H I

PROBLEM: Sensitivity Analysis for House of Fine Wines and Liquors - Chilean Wines

Parameter Values:Fixed Cost per Order: k = 50.00$ 100.00$ 150.00$ 200.00$ Annual Number of Items Demanded: A = 1,000 1,000 1,000 1,000 Unit Cost of Procuring an Item: c = 20.00$ 20.00$ 20.00$ 20.00$ Annual Holding Cost per Dollar Value: h = 0.20$ 0.20$ 0.20$ 0.20$

Decision Variables:Order Quantity: Q = 158.1 223.6 273.9 316.2

Results:Total Annual Relevant Cost: TC = 632.46$ 894.43$ 1,095.45$ 1,264.91$ Time Between Orders (years): T = 0.16 0.22 0.27 0.32

INVENTORY ANALYSIS - ECONOMIC ORDER QUANTITY MODEL

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Graphing the Sensitivity Analysis (Figure 15-7)

Sensitivity Analysis

0

200

400

600

800

1,000

1,200

1,400

$50 $100 $150 $200

Fixed Cost per Order, k

Un

its fo

r Q

* a

nd

D

olla

rs fo

r T

C(Q

*)

Order Quantity, Q*

TC(Q*)

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Backordering Model(Figure 15-9)

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101112131415161718

A B C D E F G H I J

PROBLEM: House of Fine Wines and Liquors - Chilean Wine

Parameter Values:Fixed Cost per Order: k = 100.00$ Annual Number of Items Demanded: A = 1,000 Unit Cost of Procuring an Item: c = 20.00$ Annual Holding Cost per Dollar Value: h = 0.20$ Annual Cost of Being Short One Item: p = 3.65$

Decision Variables:Economic Order Quantity: Q = 324Economic Order Level: S = 154

Results:Total Annual Relevant Cost: TC = 617.82$ Time Between Orders (years): T = 0.32

INVENTORY ANALYSIS - ECONOMIC ORDER QUANTITY MODEL WITH BACKORDERING

13

14

F

=SQRT((2*$F$7*$F$6)/($F$9*$F$8))*SQRT(($F$10+$F$9*$F$8)/$F$10)

=SQRT((2*$F$7*$F$6)/($F$9*$F$8))*SQRT($F$10/($F$10+$F$9*$F$8))

13

14

F

13

14

F

=SQRT((2*$F$7*$F$6)/($F$9*$F$8))*SQRT(($F$10+$F$9*$F$8)/$F$10)

=SQRT((2*$F$7*$F$6)/($F$9*$F$8))*SQRT($F$10/($F$10+$F$9*$F$8))

1718

F=($F$7/$F$13)*$F$6+$F$9*$F$8*(($F$14^2)/(2*F13))+((F10*(F13-F14)^2/(2*F13)))=F13/F7

17

F=($F$7/$F$13)*$F$6+$F$9*$F$8*(($F$14^2)/(2*F13))+((F10*(F13-F14)^2/(2*F13)))

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Production Model (Figure 15-13)

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101112131415161718

A B C D E F G H I

PROBLEM: Lambda Optics

Parameter Values:Fixed Set-Up Cost per Run: k = 5,000.00$ Annual Number of Items Demanded: A = 100,000 Annual Production Rate: B = 200,000 Variable Production Cost per Unit: c = 10.00$ Annual Holding Cost per Dollar Value: h = 0.20$

Decision Variables:Economic Production Quantity: Q = 31,623

Results:Time Between Production Runs (year): T = 0.32 Duration of Production Run (year): T1 = 0.16 Total Annual Relevant Cost: TC = 31,623$

INVENTORY ANALYSIS - ECONOMIC PRODUCTION-QUANTITY MODEL

13

F

=SQRT((2*F7*F6)/(F10*F9))*SQRT((F8)/(F8-F7))

1617

18

F=F13/F7=F13/F8

=(F7/F13)*F6+F10*F9*(F13/2)*((F8-F7)/F8)