McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.1 Table of Contents CD Chapter 18...

© The McGraw-Hill Companies, Inc., 200818.1McGraw-Hill/Irwin

Table of ContentsCD Chapter 18 (Inventory Management with Known Demand)

A Case Study—The Atlantic Coast Tire Corp. (ACT) Problem (Section 18.1) 18.2–18.4

Cost Components of Inventory Models (Sec.18.2) 18.5–18.6

The Basic Economic Order Quantity (EOQ) Model (Section 18.3) 18.7–18.11

The Optimal Inventory Policy for the Basic EOQ Model (Section 18.4) 18.12–18.16

The EOQ Model with Planned Shortages (Section 18.5) 18.17–18.22

The EOQ Model with Quantity Discounts (Section 18.6) 18.23–18.29

The EOQ Model with Gradual Replenishment (Section 18.7) 18.30–18.35


The Atlantic Coast Tire Corp. (ACT) Problem

• The Atlantic Coast Tire Corporation (ACT) is the east coast distributor of Eversafe tires, supplying 1500 retail stores and service stations

• When the inventory level of a particular size tire gets low, ACT places a large order with Eversafe to replenish the inventory.

• Shipments arrive by truck 9 working days after the placement of the order.

• Data for the 185/70 R13 size of Eversafe tires:– These tires sell at a regular rate of about 500 per month.

– Policy has been to order 1,000 tires as needed every couple months.

– The order is placed just in time to have the delivery arrive as inventory runs out.


Pattern of Inventory Levels for the 185/70 R13 Tire

Inventory level

Maximum = 1,000

Average = 500

Minimum = 0

0 2 4 6 8 10 12 Time (Months)


Cost Components of Inventory for 185/70 R13 Tires• The purchase price from Eversafe is $20 per tire.

• The administrative cost for placing an order is $115, due to the following:– A purchase order is initiated and processed.– The shipment must be received and placed into storage.– The computerized information processing system must be updated.

• The annual cost of holding tires in inventory is $4.20 per tire. This includes:– The cost of capital tied up in inventory (estimated at 15% of cost per annum)– The cost of leasing warehouse space.– The cost of insurance against loss by fire, theft, vandalism, etc.– The cost of personnel who oversee and protect the inventory.– Taxes that are based on the value of inventory.

• The annual cost of being out of stock is $7.50 per tire short, based on these consequences:

– Customer dissatisfaction– Potential price drop to placate a customer due to late deliveries.– Delayed revenue– The cost of additional record keeping required for out-of-stock tires.


Cost Components of Inventory Models

• Acquisition Cost. The direct cost (c) of replenishing inventory, whether through purchasing or manufacturing of the product.

– For the ACT Example: c = $20 per tire.

• Setup Cost. When purchasing the product, this cost consists of various administrative costs associated with initiating and processing the purchase order, receiving the shipment, and processing the payment. When a manufacturer is replenishing its inventory by manufacturing more of the product, the setup cost consists of the cost of setting up the manufacturing process for another production run.

– For the ACT Example: K = $115 per order.

• Holding Cost. The cost of holding units in inventory, including the cost of capital tied up in invenoty as well as the cost of space, insurance, protection, and taxes attributed to storage.

– For the ACT Example, h = $4.20 per unit per year.

• Shortage Cost. The cost incurred when there is a need for product, but none available in inventory. Possible consequences include lost sales, lost future sales, etc.

– For the ACT Example, p = $7.50 per unit short


Combining the Cost Components

• Annual acquisition cost = c times number of units added to inventory per year.

• Annual Setup cost = K times number of setups per year.

• Annual holding cost = h times average number of units in inventory.

• Annual shortage cost = p times average number of units short throughout year.

TC = total inventory cost per year = sum of above four annual costs.

TVC = total variable inventory cost per year = sum of the variable annual costs.


The Basic Economic Order Quantity (EOQ) Model

• A constant demand rate.

• The order quantity to replenish inventory arrives all at once just when desired.

• Planned shortages are not allowed.

D = annual demand rate.

Q = order quantity (the decision to be made).

Assumptions:


Reorder Point for 185/70 R13 Tires

Reorder point = (24 tires/day) (9 days) = 216 tires.


Pattern of Inventory Levels for the EOQ Model


Total Variable Cost for the 185/70 R13 Tire

Current policy: Q = 1000 tires.

Number of setups (order placements) per year is D/Q = 6000/1000 = 6.

Average inventory = Q/2 = 500.

TVC = annual setup cost + annual holding cost= 6K + 500h= 6 ($115) + 500 ($4.20)= $2,790


Optimal Solution with Different Unit Profits

Unit Profit

Doors Windows Objective Function Optimal Solution

$400 $400 Profit = 400D + 400W (2, 6)

$500 $300 Profit = 500D + 300W (4, 3)

$300 –$100 Profit = 300D – 100W (4, 0)

–$100 $500 Profit = –100D + 500W (0, 6)

–$100 –$100 Profit = –100D – 100W (0, 0)


Spreadsheet Analysis with Q = 1000

Range Name Cell

AnnualHoldingCost G7

AnnualSetupCost G6

D C4

h C6

K C5

L C7

Q C11

ReorderPoint G4

TotalVariableCost G8

WD C8

4

5

6

7

8

F G

Reorder Point = =D*(L/WD)

Annual Setup Cost = =K*(D/Q)

Annual Holding Cost = =h*(Q/2)

Total Variable Cost = =AnnualSetupCost+AnnualHoldingCost

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A B C D E F GBasic EOQ Model for Atlantic Coast Tire (Before Solving)

Data ResultsD = 6000 (demand/year) Reorder Point 216K = $115 (setup cost)h = $4.20 (unit holding cost) Annual Setup Cost $690L = 9 (lead time in days) Annual Holding Cost $2,100

WD = 250 (working days/year) Total Variable Cost $2,790

DecisionQ = 1000


Data Table (Cost vs. Order Quantity)

17

18

19

20

C D E F

Setup Holding Total

Cost Cost Cost

=AnnualSetupCost =AnnualHoldingCost =TotalVariableCost

=TABLE(,C11) =TABLE(,C11) =TABLE(,C11)

Range Name Cell


AnnualSetupCost G6

Q C11


15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

A B C D E F G H I J

Data Table for Atlantic Coast Tire Corp. (Cost vs. Order Quantity)

Order Setup Holding Total

Quantity Cost Cost Cost

1000 $690 $2,100 $2,790

100 $6,900 $210 $7,110

200 $3,450 $420 $3,870

300 $2,300 $630 $2,930

400 $1,725 $840 $2,565

500 $1,380 $1,050 $2,430

600 $1,150 $1,260 $2,410

700 $986 $1,470 $2,456

800 $863 $1,680 $2,543

900 $767 $1,890 $2,657

1000 $690 $2,100 $2,790

$0

$2,000

$4,000

$6,000

$8,000

0 200 400 600 800 1000

Order Quantity

Cost

Setup Cost Holding Cost

Total Cost

Select these

cells

(B19:E29),

before

choosing

Table from

the Data

menu.


Using Solver to Minimize Cost

Range Name Cell

D C4

h C6

HoldingCost G7

K C5

L C7

Q C11

ReorderPoint G4

SetupCost G6

TotalCost G8

WD C8

4

5

6

7

8

F G

Reorder Point =D*(L/WD)

Annual Setup Cost =K*(D/Q)

Annual Holding Cost =h*(Q/2)

Total Variable Cost =AnnualSetupCost+AnnualHoldingCost

1234567891011

A B C D E F G

Basic EOQ Model for Atlantic Coast Tire (After Solving)

Data ResultsD = 6000 (demand/year) Reorder Point 216K = $115 (setup cost)h = $4.20 (unit holding cost) Annual Setup Cost $1,204L = 9 (lead time in days) Annual Holding Cost $1,204

WD = 250 (working days/year) Total Variable Cost $2,407

DecisionQ = 573.21


The Square Root Formula

Number of setups per year = D/Q.

Average inventory level = Q/2.

Total variable cost (TVC) =

The value of Q that minimizes the TVC is

For ACT’s problem,

Which gives TVC = $115 (6,000 / 573) + $4.20 (573 / 2) = $2407, a 14% savings over current policy.

KD

Q

⎛⎝⎜

⎞⎠⎟+h

Q2

⎛⎝⎜

⎞⎠⎟

Q* =2KD

h

Q* =2(115)(6,000)

4.20=573.


Sensitivity Analysis for ACT

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

A B C D E F G H I J K L

Sensitivity Analysis for Atlantic Coast Tire Corp.

Unit Holding Cost

573 $3.78 $3.99 $4.20 $4.41 $4.62

$103.50 573 558 544 531 518

Setup $109.25 589 573 559 545 533

Cost $115.00 604 588 573 559 547

$120.75 619 603 587 573 560

$126.50 634 617 601 587 573

Optimal Order Quantity

Unit Holding Cost

$2,407 $3.78 $3.99 $4.20 $4.41 $4.62

$103.50 $2,167 $2,226 $2,284 $2,340 $2,395

Setup $109.25 $2,226 $2,287 $2,347 $2,404 $2,461

Cost $115.00 $2,284 $2,347 $2,407 $2,467 $2,525

$120.75 $2,340 $2,404 $2,467 $2,528 $2,587

$126.50 $2,395 $2,461 $2,525 $2,587 $2,648

Total Variable Cost (with Q = Q*)

Unit Holding Cost

$2,407 $3.78 $3.99 $4.20 $4.41 $4.62

$103.50 $2,167 $2,227 $2,287 $2,347 $2,407

Setup $109.25 $2,227 $2,287 $2,347 $2,407 $2,468

Cost $115.00 $2,287 $2,347 $2,407 $2,468 $2,528

$120.75 $2,347 $2,408 $2,468 $2,528 $2,588

$126.50 $2,408 $2,468 $2,528 $2,588 $2,648

Total Variable Cost (with Q=573)

Select the entire table

(C18:H23), before

choosing Table from

the Data menu.

Select the entire table

(C28:H33), before

choosing Table from

the Data menu.

Replace square root

formula for Q (cell

C11) with 573. Select

the entire table

(C38:H43), before

choosing Table from

the Data menu.

Range Name Cell

h C6

K C5

Q C11


18

19

C D

=Q 3.78

103.5 =TABLE(C6,C5)

28

29

C D

=TotalVariableCost 3.78

103.5 =TABLE(C6,C5)

38

39

C D

=TotalVariableCost 3.78

103.5 =TABLE(C6,C5)


The EOQ Model with Planned Shortages


• The order quantity to replenish inventory arrives all at once just when desired.

• Planned shortages are allowed. When a shortage occurs, the affected customers will wait for the product to become available again. Their backorders are filled immediately when the order quantity arrives.

Q = order quantity.

S = maximum shortage (units backordered).

Assumptions:

Decision Variables:


Pattern of Inventory Level with Planned Shortages


The Objective of the Model: Minimize TVC

Annual Setup cost =

Annual holding cost = h times (average inventory level when positive) times (fraction of time inventory level is positive)

Annual shortage cost = p times (average shortage level when a shortage occurs) times (fraction of time shortage is occurring)

Combining these gives

KD

Q

⎛⎝⎜

⎞⎠⎟

=hQ − S

2⎛⎝⎜

⎞⎠⎟

Q − S

Q

⎛⎝⎜

⎞⎠⎟

= h(Q − S)2

2Q

=pS

2⎛⎝⎜

⎞⎠⎟

S

Q

⎛⎝⎜

⎞⎠⎟

= pS2

2Q

TVC =KDQ

⎛⎝⎜

⎞⎠⎟+h

(Q−S)2

2Q+ p

S2

2Q


The Optimal Inventory Policy

Q* =h+ p

p2KD

h

S* =h

h+ p⎛⎝⎜

⎞⎠⎟Q*

whereD = annual demand rate,K = setup cost,h = unit holding cost,p = unit shortage cost

Maximum inventory level = Q* – S* =p

h + p

2KD

h


Application to the ACT Case Study

1011

B CQ = =SQRT(2*D*K/h)*SQRT((p+h)/p)S = =(h/(h+p))*Q

Range Name Cell


AnnualSetupCost G6

AnnualShortageCost G8

D C4

h C6

K C5

MaxInventoryLevel G4

p C7

Q C10

S C11


456789

F GMax Inventory Level =Q-S

Annual Setup Cost =K*D/QAnnual Holding Cost =h*(MaxInventoryLevel^2)/(2*Q)

Annual Shortage Cost =p*((Q-MaxInventoryLevel)^2)/(2*Q)Total Variable Cost =AnnualSetupCost+AnnualHoldingCost+AnnualShortageCost

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1011

A B C D E F GEOQ Model with Planned Shortages for Atlantic Coast Tire

Data ResultsD = 6000 (demand/year) Max Inventory Level 458.94K = $115 (setup cost)h = $4.20 (unit holding cost) Annual Setup Cost $963.77p = $7.50 (unit shortage cost) Annual Holding Cost $617.80

Annual Shortage Cost $345.97Decision Total Variable Cost $1,927.53

Q = 715.94 (order quantity)S = 257.00 (maximum shortage)

Solver Version:

Analytical Version:


Comparison of Basic EOQ and EOQ with Planned Shortages

Quantity Basic EOQ ModelEOQ Model with

Planned Shortages

Order quantity 573 716

Maximum shortage 0 257

Maximum inventory level

573 459

Reorder point 216 –41

Annual setup cost $1,204 $964

Annual holding cost $1,204 $618

Annual shortage cost 0 $346

Total variable cost $2,407 $1,928


The Quantity Discounts Being Offered to ACT

Discount Quantity

OrderQuantity Discount Unit Cost

1 0 to 749 0 $20.00

2 750 to 1,999 1% $19.80

3 2,000 or more 2% $19.60


Cost Analysis

TVC =

wherec = unit acquisition cost (as given in Table 18.2)D = annual demand rate = 6,000K = setup cost = $115Q = order quantity (the decision variable)h = unit holding cost

I = inventory holding cost rate = 0.21.h = Ic = 0.21c

cD + KDQ

⎛⎝⎜

⎞⎠⎟+h

Q2

⎛⎝⎜

⎞⎠⎟


The Unit Holding Cost for ACT

DiscountCategory

Price(c)

Unit Holding Costh = Ic = 0.21c

1 $20.00 0.21($20) = $4.20

2 $19.80 0.21($19.80) = $4.158

3 $19.60 0.21($19.60) = $4.116


Total Variable Cost vs. Order Quantity


Cost Comparison for Discount Categories

Annual Costs

DiscountCategory

Best OrderQuantity

Acquisition6,000c

Setups$115(6000/Q)

Holdingh(Q/2)

Total (TVC)= Sum

1 Q = 573 $120,000 $1,204 $1,204 $122,407

2 Q = 750 $118,800 $920 $1,559 $121,279

3 Q = 2,000 $117,600 $345 $4,116 $122,061


Spreadsheet Analysis with Quantity Discounts

123456789101112131415161718192021

A B C D E F G H I J KEOQ Model with Quantity Discounts for Atlantic Coast Tire Corp.

DataD = 6000 (demand/year)K = $115 (setup cost)I = 0.21 (inventory holding cost rate)

N = 3 (number of discount categories)

Annual Annual Annual TotalRange of order quantities Purchase Setup Holding Variable

Category Price Lower LimitUpper LimitEOQ Q* Cost Cost Cost Cost1 $20.00 0 749 573 573 $120,000 $1,204 $1,204 $122,4072 $19.80 750 1999 576 750 $118,800 $920 $1,559 $121,2793 $19.60 2000 10000000 579 2000 $117,600 $345 $4,116 $122,061

ResultsOptimal Q 750

Total Variable Cost $121,279


Conclusion of the ACT Cast Study

• A substantial reduction in the order quantity (from the current 1,000 down to 750) would provide a significant reduction in average inventory level and holding cost.

• The threat to reduce the order quantity even further (as suggested by the basic EOQ model) has prodded Eversafe to provide quantity discounts to ACT.

• The resulting reduction in the total annual inventory cost from that for the current policy would exceed $1,5000 for just this one size of tire. Extending this approach to the other sizes should greatly multiply this savings.


The EOQ Model with Gradual Replenishment


• A production run is scheduled to begin each time the inventory level drops to 0, and this production replenishes inventory at a constant rate throughout the duration of the run.

• Planned shortages are not allowed.

Q = production lot size.

Assumptions:

Decision Variable:


Pattern of Inventory Level with Gradual Replenishment


The SOCA Problem

• SOCA, a television manufacturing company, produces its own speakers for assembly into television sets.

• Current policy for managing SOCA’s inventory of speakers:– Daily demand rate = 1,000 speakers per day.

– Daily production rate = 3,000 speakers per day (when producing).

– The production facilities get set up to start a production run each time the inventory level is scheduled to drop to 0.

– Each production run produces 30,000 speakers over a period of 10 working days, so another 20 working days elapse before the next production run is needed.


TVC for the Current Policy

Maximum inventory level = production lot size minus demand during production run= 30,000 – (10 days)(1,000 speakers / day)= 20,000 speakers

Average inventory level = (1/2) (maximum inventory level) = 10,000 speakers

Annual setup cost

Annual holding cost

TVC = annual setup cost + annual holding cost = $136,000.

=KD

Q

⎛⎝⎜

⎞⎠⎟

= ($12,000 / setup)250,000 speakers

30,000 speakers / setup

= $100,000

=h(average inventory level)

= ($3.60 / speaker)(10,000 speakers)

= $36,000


The Optimal Inventory Policy

Q* =2KD

h 1−DR

⎛⎝⎜

⎞⎠⎟

whereD = annual demand rate,R = annual production rate (if produce continuously)K = setup cost,h = unit holding cost,

For SOCA, Q* =2(12,000)(250,000)

3.60 1−250,000750,000

⎛⎝⎜

⎞⎠⎟

=50,000,

TVC =$12,000250,00050,000

+$3.60(25,000) 1−250,000750,000

⎛⎝⎜

⎞⎠⎟=$120,000.


Spreadsheet Analysis with Gradual Replenishment

Range Name CellAnnualHoldingCost G4AnnualShortageCost G5

D C4

h C7K C6PR C5Q C10TotalVariableCost G6

456

F GAnnual Holding Cost =K*D/Q

Annual Shortage Cost =h*(1-D/PR)*Q/2Total Variable Cost =AnnualHoldingCost+AnnualShortageCost

10B C

Q = =SQRT(2*D*K/(h*(1-D/PR)))

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10

A B C D E F G

EOQ Model with Gradual Replenishment for SOCA

Data ResultsD = 250,000 (demand/year) Annual Holding Cost $60,000

PR = 750,000 (production rate) Annual Shortage Cost $60,000K = $12,000 (unit setup cost) Total Variable Cost $120,000h = $3.60 (unit holding cost)

DecisionQ = 50,000 (production lot size)

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.1 Table of Contents CD Chapter 18...

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Transcript of McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 18.1 Table of Contents CD Chapter 18...