1 chapter 15 if i order too little, i make no profit. if i order too much, i may go broke. every...

1

Chapter 15

If I order too little, I make no profit. If I order too much, I may go broke. Every product is different. Help me!—A Retailer’s Plea

Inventory Decisions

with Certain Factors

2

Elements of Inventory Decisions

There are four basic inventory system costs: Ordering costs Procurement costs Inventory holding or carrying costs Inventory shortage costs

Demand is usually erratic and uncertain. We assume it is smooth and predictable. That makes it easier to develop mathematical

models. These can later be made more realistic. Order quantity is the main variable.

With no uncertainty, we can schedule deliveries to arrive exactly when we run out.

3

The Economic Order Quantity(EOQ) Model

The decision variable isQ = Order Quantity

There are four parameters:k = Fixed cost per orderA = Annual number of items

demandedc = Unit cost of procuring an itemh = Annual cost per dollar value of holding items in inventory

An order quantity is to be found that minimizes:Total

=Ordering

+Holding

+Procurement

Annual cost Cost Cost Cost

4


Inventory level has a cycle beginning with a new shipment’s arrival.

T = Q/A = Duration of inventory cycle

5


The annual ordering cost is the number of orders times the cost per order:

The annual holding cost is the cost per item held 1year 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

6


The total annual inventory cost is:

We drop Ac from the above, since that amount will not vary with Q. Ac is not a relevant cost.

That provides the function to be minimized, the total annual relevant inventory cost:

AcQ

hckQA

2 cost annual Total

2 )(

Qhck

QA

QTC

7


It may be shown using calculus that the level for Q minimizing the above is the economic order quantity

Problem. A software store sells 500 Alien Saboteurs annually. The supplier charges $100 per order plus $20 each. It costs $.15 per dollar value to hold inventory for a year. How many should they order, how often, and at what annual relevant inventory cost?

hcAk

*Q2

8


Solution: The following parameters apply:

A = 500 k = 100 c = 20 h = .15 The economic order quantity is

The inventory cycle duration is T = Q/A = 183/500 = .366 year or 133.6 days

The total annual relevant inventory cost is:

183or6182201510050022

..hc

Ak*Q

72547$5027422273$2

1832015100

183500

(183) ...)(.TC

9

Optimal Inventory Policywith Backordering

Retailers may not stock all demand. Orders placed during shortages are backordered.

10


The new model adds the order level S, that quantity on hand when a shipment arrives.

A shortage cost applies, based on a penalty p for being one item short for a year.

New total annual relevant inventory cost:

Optimal order quantity and order level:

Q

SQpQ

hcSk

QA

S,QTC22

)(22

hcpp

hcAk

*Sp

hcphcAk

*Q

2

2

11


Shortage penalty p applies over a year, but cost prorates to fractions of items or years.

Example: The retailer suffers lost profit on future business equal to $.05 each day that one Alien Saboteur is on backorder. That translates into p = $.05×365 = $18.25.

Solution: The order quantity is computed:

251820152518

201510050022

...

.phcp

hcAk

*Q

197or04197

251820152518

6182 ....

.

12


Solution: The order level is computed:

The relevant cost is

= $253.81 + 217.47 + 36.31 = $507.59 The above is smaller than before, even though

there is a shortage penalty and shortages. Why?

20152518

2518201510050022

..

..hcp

phcAk

*S

169or22169

201525182518

6182 ...

..

1972

16919725181972

1692015100

197500

)169197(22

..

,TC

13


There is a net savings in holding costs and a slight reduction in ordering costs. Those outweigh increased cost due to shortages.

The number of backorders is Q – S. Here that quantity is 197 – 169 = 28.

The annual shortage cost is only $36.31, because durations of shortage (for last of the 28) are only 28/197 = .142 year (52 days).

The results suggest that: Retailers will run short, if they can get away

with it! But backordering must make sense.

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Nobody backorders cigarettes or gasoline. Sales for those products are lost during

shortages. This model does not apply for them. The shortage penalty p is not usually

known. But it may be imputed from existing policy. The service level L is used for that purpose:

L = proportion of time fully stocked The imputed shortage penalty is:

p =hcL1 L

15

Economic Production-Quantity Model

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

16


The new parameter is the annual production rate B.

Parameter k is the production setup cost. The variable production cost per unit is c. The total annual relevant inventory cost:

The economic production quantity:

BABQ

hckQA

QTC2

)(

BAB

hcAk

*Q 2

17


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

Economic production quantity is

Total relevant cost is TC(8,944) 5651629

000500000100000500

29448

54000029448000100

.,,

,,,.,

,,

thousand9448500

100500540

210022 .

.BAB

hcAk

*Q

18

More Elaborate Models

Incorporate a second one-time shortage penalty (done in Chapter 16).

These models are for single products. Add additional products.

Incorporate uncertainty regarding: Demand (done in Chapter 16). Lead-time for delivery of order (Chapter 16).

Incorporate lost sales (done in Chapter 16). Extend to single period products (Ch. 16). NOTE: The basic EOQ model works very

well even when its ideal conditions don’t apply. It is very robust.

19

Inventory Spreadsheet Templates

Economic Order Quantity

Sensitivity Analysis

Backordering

Production

20

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

1. Enter the problem name in B3.


2. Enter the problem information in F6:F9.


Optimal order quantity

Optimal order quantity

Optimal total annual relevant cost and time between orders


21

Sensitivity Analysis(Figure 15-6)



2. Enter the problem information in F6:I9.

2. Enter the problem information in F6:I9.

A sensitivity analysis shows how answers vary as data changes. Here the fixed order cost, k, varies.

12345678910111213141516

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

The fixed order cost has a diminishing effect on the results. For example, a 100% increase in k causes both Q* and TC(Q)* to increase by only 41%.

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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*)

Graphing sensitivity analysis results makes It is easier to see relationships.

Graphing sensitivity analysis results makes It is easier to see relationships.

23

Backordering Model(Figure 15-9)

123456789

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

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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)))





Optimal order quantity and order level

Optimal order quantity and order level



24

Production Model (Figure 15-13)

123456789

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)


1. Enter the problem name in B3. 2. Enter the problem

information in F6:F10.


Optimal order quantity Optimal order quantity Optimal time between production runs, duration of production run, and total annual relevant cost.

Optimal time between production runs, duration of production run, and total annual relevant cost.

1 chapter 15 if i order too little, i make no profit. if i order too much, i may go broke. every...

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