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 Aslı Sencer

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? Aslı Sencer

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 houses Detergent consumption in houses Deterministic (certain) versus stochastic (uncertain) demand Ex: 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 demand Ex: Demand quantities for the next months are 20, 21, 20, 19 Demand quantities for the next months are 20, 50, 10, 2 Aslı Sencer

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 Quantity EOQ-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 Annual cost = Ordering Cost + Holding Cost + Procurement Cost Aslı Sencer

EOQ Inventory Policy

Average Inv. Level Aslı Sencer

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: Annual ordering cost   

A Q

 

k

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

hc Q

2  The annual procurement cost is the product of annual demand and unit cost: Procurement cost =

Ac

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Annual Cost of EOQ-Model

 Total annual cost   

A Q

 

k



hc

 

Q

2 

Ac

Here

Ac

is not a relevant cost and thus dropped.

 Minimize Total Annual Inventory Cost :

TC

(

Q

)   

A Q

 

k



hc Q

2 Aslı Sencer

Optimal Solution of EOQ

 Optimal solution is the economic order quantity

Q

*



2

Ak hc

 Optimal Total Cost

TC

*



2

Akhc

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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 rate

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

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Solution:

The economic order quantity is

Q

*  2

Ak



hc

2  5200 .

20 2      509 .

9 or 510   The inventory cycle duration is

T

=

Q/A

= 510/5200 = 0.098 year or 36 days The total annual relevant inventory cost is:

TC

(510)    5200 510   10  .

20 ( 2 ) 510 2  $ 101 .

96  102 .

00  $ 203 .

96 /

year

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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

A actual

=

4 A estimated

Then

EOQ actual

=

2 A estimated

Since

EOQ actual

 2

A actual k hc

 2 2

A estimated k hc

 2

EOQ estimated

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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:

Q

*  2

Ak



hc

2  1000 .

20      223 .

6 or 224

T

=

Q/A

= 24/1000 = .224 year or 82 days

TC

(224)    1000 224   100  .

20 ( 20 ) 224 2  $894.43/ye ar Aslı Sencer

Optimal Inventory Policy with Backordering

Orders placed during shortages are backordered.

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Optimal Inventory Policy with Backordering

S

: Quantity on hand when a shipment arrives.

P:

Cost of being one item short for a year

TC

(

Q , S

)   

A Q

 

k



hcS

2 2

Q



p



Q



S

 2 2

Q

Optimal order quantity and order level:

Q *

 2

Ak hc p



hc p S *

 2

Ak hc p



p hc

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Example:The House of Wines and Liquors-Backorders

The 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 .

Q

*  2

Ak hc p



p hc

 2  1000 .

20     Aslı Sencer 3 .

65  .

20 3 .

65  324

Example: Solution

 The order level S is

S

*  2

Ak hc p p



hc

 2  1000 .

20     3 .

65 3 .

65  .

20  154  The relevant cost is

TC

( 324 , 154 )    1000 324   100  .

20 20 2 154   2  3 .

65 2     2  617 .

82 smaller than before, why?

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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 Penalty

An 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

Q

*



S

*

Q

*



1



L

, so

LQ

*



S *

Imputed shortage penalty

p

=

hcL

1 

L

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As p increases EOQ is more robust

224 A=1000 units/yr k=$100/order c=$20/unit h= $0.20/$/year L=47.5% 324 Q* S* L=90% 236 212 EOQ with no backordering 154 $3.65

Aslı Sencer $36 imputed shortage penalty P

Economic Production-Quantity Model

The inventory model may be extended to finding the

optimal production quantity

.

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

Economic Production-Quantity

   

Model

B

: Annual production rate

K:

Production setup cost.

c

: Variable production cost per unit.

Total Annual Cost: TC

(

Q

)

  

A Q

 

k



hc

 

Q

2

   

B B A

 

Economic Production Quantity:

Q *

 2

Ak hc B



B A

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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

Q

*



2

Ak hc B



A B



2



100 .

40

   

2

Total relevant cost is

500



100 500



8 , 944 units

TC

(8,944)   

100 , 000 8 , 944

  

2 , 000

 

.

40 8 , 944 2

 

500 , 000 500



, 100 , 000 000

  

$ 29 , 516 .

56

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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 Aslı Sencer

Economic Order Quantity Model

(Figure 15-3) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A B

Results:

C

Decision Variables:

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

Parameter Values:

Fixed Cost per Order: k = Annual Number of Items Demanded: A = Unit Cost of Procuring an Item: c = Annual Holding Cost per Dollar Value: h = Order Quantity: Q = Total Annual Relevant Cost: TC = Time Between Orders (years): T = F

INVENTORY ANALYSIS - ECONOMIC ORDER QUANTITY MODEL

$ 10.00

5,200 $ 2.00

$ 0.20

100 $ 540.00

0.0192

15 16 G H I F =(F7/F12)*F6+F9*F8*(F12/2) =F12/F7 Aslı Sencer

Sensitivity Analysis

(Figure 15-6) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A B

Results:

C

Decision Variables:

D Order Quantity: Q = E F G H

INVENTORY ANALYSIS - ECONOMIC ORDER QUANTITY MODEL

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

Parameter Values:

Fixed Cost per Order: k = Annual Number of Items Demanded: A = Unit Cost of Procuring an Item: c = Annual Holding Cost per Dollar Value: h = Total Annual Relevant Cost: TC = Time Between Orders (years): T = $ $ $ $ 50.00

1,000 20.00

0.20

158.1

632.46

0.16

$ 100.00

$ $ $ 1,000 20.00

894.43

0.20

223.6

0.22

$ 150.00

1,000 $ 20.00

$ 0.20

273.9

$ 1,095.45

0.27

I $ 200.00

1,000 $ 20.00

$ 0.20

316.2

$ 1,264.91

0.32

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Graphing the Sensitivity Analysis (Figure 15-7)

Sensitivity Analysis

1,400 1,200 1,000 800 600 400 200 0 $50 $100 $150 Fixed Cost per Order, k Order Quantity, Q* TC(Q*) $200 Aslı Sencer

Backordering Model

(Figure 15-9) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 A B

Results:

C

Decision Variables:

D Economic Order Level: S = E PROBLEM: House of Fine Wines and Liquors - Chilean Wine

Parameter Values:

Fixed Cost per Order: k = Annual Number of Items Demanded: A = Unit Cost of Procuring an Item: c = Annual Holding Cost per Dollar Value: h = Annual Cost of Being Short One Item: p = Economic Order Quantity: Q = Total Annual Relevant Cost: TC = Time Between Orders (years): T = F $ 100.00

1,000 $ 20.00

$ 0.20

$ 3.65

324 154 $ 617.82

0.32

G H F I J

INVENTORY ANALYSIS - ECONOMIC ORDER QUANTITY MODEL WITH BACKORDERING

13 =SQRT((2*$F$7*$F$6)/($F$9*$F$8)) *SQRT(($F$10+$F$9*$F$8)/$F$10) 14 17 18 =SQRT((2*$F$7*$F$6)/($F$9*$F$8)) *SQRT($F$10/($F$10+$F$9*$F$8)) F =($F$7/$F$13)*$F$6+$F$9*$F$8* (($F$14^2)/(2*F13))+((F10*(F13 F14)^2/(2*F13))) =F13/F7 Aslı Sencer

Production Model

(Figure 15-13) 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 A B PROBLEM: Lambda Optics

Results:

C

Decision Variables:

D

Parameter Values:

Fixed Set-Up Cost per Run: k = Annual Number of Items Demanded: A = Annual Production Rate: B = Variable Production Cost per Unit: c = Annual Holding Cost per Dollar Value: h = E

INVENTORY ANALYSIS - ECONOMIC PRODUCTION-QUANTITY MODEL

Economic Production Quantity: Q = Time Between Production Runs (year): T = Duration of Production Run (year): T1 = Total Annual Relevant Cost: TC = F $ 5,000.00

100,000 200,000 $ 10.00

$ 0.20

31,623 0.32

0.16

$ 31,623 13 16 17 18 G H I F =SQRT((2*F7*F6)/(F10*F9))*S QRT((F8)/(F8-F7)) F =F13/F7 =F13/F8 =(F7/F13)*F6+F10*F9*(F13/2)* ((F8-F7)/F8) Aslı Sencer